Nothing
R/profile.R
In mev: Modelling of Extreme Values
Defines functions
gpd.tem
gev.tem
tem.corr
spline.corr
plot.fr
gpd.pll
gev.pll
plot.eprof
confint.eprof
Documented in
confint.eprof
gev.pll
gev.tem
gpd.pll
gpd.tem
plot.eprof
plot.fr
spline.corr
tem.corr
# This function is added to ensure that the local solver can be fixed to something different
# than COBYLA - see https://github.com/jyypma/nloptr/pull/38 and
# https://github.com/stevengj/nlopt/issues/118
# .auglag <- function(x0, fn, gr = NULL, lower = NULL, upper = NULL, hin = NULL, hinjac = NULL,
# heq = NULL, heqjac = NULL, localsolver = c("COBYLA"), localtol = 1e-06, ineq2local = FALSE,
# nl.info = FALSE, control = list(), ...) {
# if (ineq2local) {
# stop("Inequalities to local solver: feature not yet implemented.")
# }
# localsolver <- toupper(localsolver)
# if (localsolver %in% c("COBYLA", "BOBYQA")) {
# # changed this line to add BOBYQA local solver
# dfree <- TRUE
# gsolver <- "NLOPT_LN_AUGLAG"
# lsolver <- paste("NLOPT_LN_", localsolver, sep = "")
# } else if (localsolver %in% c("LBFGS", "MMA", "SLSQP")) {
# dfree <- FALSE
# gsolver <- "NLOPT_LD_AUGLAG"
# lsolver <- paste("NLOPT_LD_", localsolver, sep = "")
# } else {
# stop("Only local solvers allowed: BOBYQA, COBYLA, LBFGS, MMA, SLSQP.")
# }
# .fn <- match.fun(fn)
# fn <- function(x) .fn(x, ...)
# if (!dfree && is.null(gr)) {
# gr <- function(x) nloptr::nl.grad(x, fn)
# }
# opts <- nloptr::nl.opts(control)
# opts$algorithm <- gsolver
# local_opts <- list(algorithm = lsolver, xtol_rel = localtol, eval_grad_f = if (!dfree) gr else NULL)
# opts$local_opts <- local_opts
# if (!is.null(hin)) {
# .hin <- match.fun(hin)
# hin <- function(x) (-1) * .hin(x)
# }
# if (!dfree) {
# if (is.null(hinjac)) {
# hinjac <- function(x) nloptr::nl.jacobian(x, hin)
# } else {
# .hinjac <- match.fun(hinjac)
# hinjac <- function(x) (-1) * .hinjac(x)
# }
# }
# if (!is.null(heq)) {
# .heq <- match.fun(heq)
# heq <- function(x) .heq(x)
# }
# if (!dfree) {
# if (is.null(heqjac)) {
# heqjac <- function(x) nloptr::nl.jacobian(x, heq)
# } else {
# .heqjac <- match.fun(heqjac)
# heqjac <- function(x) .heqjac(x)
# }
# }
# S0 <- nloptr::nloptr(x0, eval_f = fn, eval_grad_f = gr, lb = lower, ub = upper, eval_g_ineq = hin,
# eval_jac_g_ineq = hinjac, eval_g_eq = heq, eval_jac_g_eq = heqjac, opts = opts)
# if (nl.info)
# print(S0)
# S1 <- list(par = S0$solution, value = S0$objective, iter = S0$iterations, global_solver = gsolver,
# local_solver = lsolver, convergence = S0$status, message = S0$message)
# return(S1)
# }
#' Confidence intervals for profile likelihood objects
#'
#' Computes confidence intervals for the parameter psi for profile likelihood objects.
#' This function uses spline interpolation to derive \code{level} confidence intervals
#'
#' @param object an object of class \code{eprof}, normally the output of \link{gpd.pll} or \link{gev.pll}.
#' @param parm a specification of which parameters are to be given confidence intervals,
#' either a vector of numbers or a vector of names. If missing, all parameters are considered.
#' @param level confidence level, with default value of 0.95
#' @param prob percentiles, with default giving symmetric 95\% confidence intervals
#' @param method string for the method, either \code{cobs} (constrained robust B-spline from eponym package) or \code{smooth.spline}
#' @param ... additional arguments passed to functions. Providing a logical \code{warn=FALSE} turns off warning messages when the lower or upper confidence interval for \code{psi} are extrapolated beyond the provided calculations.
#' @param print should a summary be printed. Default to \code{FALSE}.
#' @return returns a 2 by 3 matrix containing point estimates, lower and upper confidence intervals based on the likelihood root and modified version thereof
#' @export
#' @importFrom Rsolnp solnp
#' @importFrom alabama auglag
#' @importFrom utils tail
confint.eprof <- function(object, parm, level = 0.95, prob = c((1-level)/2, 1-(1-level)/2), print = FALSE, method = c("cobs","smooth.spline"), ...) {
if(!isTRUE(all.equal(diff(prob),level, check.attributes = FALSE))){
warning("Incompatible arguments: \"level\" does not match \"prob\".")
}
method <- match.arg(method[1], c("cobs","smooth.spline"))
args <- list(...)
if ("warn" %in% names(args) && is.logical(args$warn)) {
warn <- args$warn
} else {
warn <- TRUE
}
if (length(prob) != 2) {
stop("\"prob\" must be a vector of size 2")
prob <- sort(prob)
}
if (missing(parm)) {
parm <- NULL
ind <- args$ind
if (!is.null(object$pll) || !is.null(object$r)) {
parm <- c(parm, "profile")
ind <- c(ind, 1)
}
if (!is.null(object$rstar)) {
parm <- c(parm, "tem")
ind <- c(ind, 2)
}
if (!is.null(object$tem.pll)) {
parm <- c(parm, "modif.tem")
ind <- c(ind, 3)
}
if (!is.null(object$empcov.pll)) {
parm <- c(parm, "modif.empcov")
ind <- c(ind, 4)
}
} else {
if (is.numeric(parm)) {
ind <- parm
parm <- c("profile", "tem", "modif.tem", "modif.empcov")[ind]
} else {
parm <- match.arg(arg = parm, choices = c("profile", "tem", "modif.tem", "modif.empcov",
"r", "rstar"), several.ok = TRUE)
parm[parm %in% "r"] <- "profile"
parm[parm %in% "rstar"] <- "tem"
ind <- which(c("profile", "tem", "modif.tem", "modif.empcov") %in% parm)
}
parm <- unique(parm)
ind <- unique(ind[ind %in% 1:4])
}
if (length(ind) == 0) {
stop("Invalid \"parm\" argument.")
}
qulev <- qnorm(1-prob)
conf <- matrix(ncol = 4, nrow = 3)
for (i in ind) {
if (i == 1) {
if (is.null(object$pll) && is.null(object$r)) {
break
}
if (is.null(object$r)) {
# no r object, but must have pll + maxpll
object$r <- sign(object$psi.max - object$psi) * sqrt(2 * (object$maxpll - object$pll))
} else{
object$r[is.infinite(object$r)] <- NA
}
if (is.null(object$normal)) {
object$normal <- c(object$psi.max, object$std.error)
}
if (method == "cobs" && requireNamespace("cobs", quietly = TRUE)) {
fit.r <- cobs::cobs(x = object$r, y = object$psi, constraint = "decrease",
lambda = 0, ic = "SIC", pointwise = cbind(0, 0, object$normal[1]), knots.add = TRUE,
repeat.delete.add = TRUE, print.mesg = FALSE, print.warn = FALSE)
pr <- predict(fit.r, c(0, qulev))[,2]
} else {
fit.r <- stats::smooth.spline(x = na.omit(cbind(object$r, object$psi)), cv = FALSE)
pr <- predict(fit.r, c(0, qulev))$y
pr[1] <- object$normal[1]
}
conf[,i] <- pr
if (warn) {
if (!any(object$r > qnorm(prob[1]))) {
warning("Extrapolating the lower confidence interval for the profile likelihood ratio test")
}
if (!any(object$r < qnorm(prob[2]))) {
warning("Extrapolating the upper confidence interval for the profile likelihood ratio test")
}
}
} else if (i == 2) {
if (is.null(object$rstar)) {
break
}
if (method == "cobs" && requireNamespace("cobs", quietly = TRUE)) {
fit.rst <- cobs::cobs(x = object$rstar, y = object$psi, constraint = "decrease",
lambda = 0, ic = "SIC", knots.add = TRUE, repeat.delete.add = TRUE, print.mesg = FALSE,
print.warn = FALSE)
prst <- predict(fit.rst, c(0, qulev))[,2]
} else {
fit.rst <- stats::smooth.spline(x = na.omit(cbind(object$rstar, object$psi)),
cv = FALSE)
prst <- predict(fit.rst, c(0, qulev))$y
}
if (!is.null(object$tem.psimax)) {
prst[1] <- object$tem.psimax
} else{
object$tem.psimax <- prst[1]
}
conf[,i] <- prst
# lines(x=object$rstar,fit.rst$fitted,col=2,pch=19)
if (warn) {
if (!any(object$rstar > qulev[2])) {
warning("Extrapolating the adjusted lower confidence interval for rstar.")
}
if (!any(object$rstar < qulev[1])) {
warning("Extrapolating the adjusted upper confidence interval for rstar")
}
}
} else if (i == 3) {
if (is.null(object$tem.pll)) {
break
}
if (method == "cobs" && requireNamespace("cobs", quietly = TRUE)) {
fit.mtem <- cobs::cobs(x = sign(object$tem.mle - object$psi) * sqrt(-2 * (object$tem.pll -
object$tem.maxpll)), y = object$psi, constraint = "decrease", lambda = 0,
ic = "SIC", knots.add = TRUE, repeat.delete.add = TRUE, print.mesg = FALSE,
print.warn = FALSE)
ptem <- predict(fit.mtem, c(0, qulev))[, 2]
} else {
fit.mtem <- stats::smooth.spline(x = na.omit(cbind(sign(object$tem.mle - object$psi) *
sqrt(-2 * (object$tem.pll - object$tem.maxpll)), object$psi)), cv = FALSE)
ptem <- predict(fit.mtem, c(0, qulev))$y
}
ptem[1] <- object$tem.mle
conf[,i] <- ptem
#TODO add warnings about extrapolation
} else if (i == 4) {
if (is.null(object$empcov.pll)) {
break
}
if (method == "cobs" && requireNamespace("cobs", quietly = TRUE)) {
fit.mempcov <- cobs::cobs(x = sign(object$empcov.mle - object$psi) * sqrt(-2 *
(object$empcov.pll - object$empcov.maxpll)), y = object$psi, constraint = "decrease",
lambda = 0, ic = "SIC", knots.add = TRUE, repeat.delete.add = TRUE, print.mesg = FALSE,
print.warn = FALSE)
pempcov <- predict(fit.mempcov, c(0.5, qulev))[, 2]
} else {
fit.mempcov <- stats::smooth.spline(x = na.omit(cbind(sign(object$empcov.mle -
object$psi) * sqrt(-2 * (object$empcov.pll - object$empcov.maxpll)), object$psi)),
cv = FALSE)
pempcov <- predict(fit.mempcov, c(0.5, qulev))$y
}
pempcov[1] <- object$empcov.mle
conf[,i] <- pempcov
}
}
if (!is.null(conf)) {
colnames(conf) <- c("Profile", "TEM", "Severini (TEM)", "Severini (emp. cov.)")
rownames(conf) <- c("Estimate", "Lower CI", "Upper CI")
#Check output makes sense - lower CI is lower than estimate
#and similarly upper CI is above estimate
wrong_below <- which(conf[2,] > conf[1,])
if(length(wrong_below)>0){
conf[2,][wrong_below] <- NA
}
wrong_above <- which(conf[3,] < conf[1,])
if(length(wrong_above)>0){
conf[3,][wrong_above] <- NA
}
if(print){
cat("Point estimate for the parameter of interest psi:\n")
cat("Maximum likelihood :", round(object$psi.max, 3), "\n")
if(2 %in% ind){
cat("Tangent exponential model :", round(object$tem.psimax, 3), "\n")
}
if(3 %in% ind){
cat("Severini's profile (TEM) :", round(object$tem.mle, 3), "\n")
}
if(4 %in% ind){
cat("Severini's profile (empcov) :", round(object$empcov.mle, 3), "\n")
}
cat("\n")
cat("Confidence intervals, levels :", prob, "\n")
cat("Wald intervals :", round(object$psi.max+sort(qulev)*object$std.error, 3), "\n")
cat("Profile likelihood :", round(conf[2:3,1], 3), "\n")
if(2 %in% ind){
cat("Tangent exponential model :", round(conf[2:3,2], 3), "\n")
}
if(3 %in% ind){
cat("Severini's profile (TEM) :", round(conf[2:3,3], 3), "\n")
}
if(4 %in% ind){
cat("Severini's profile (empcov) :", round(conf[2:3,4], 3), "\n")
}
}
return(invisible(conf[,ind]))
}
}
#' Plot of (modified) profile likelihood
#'
#' The function plots the (modified) profile likelihood and the tangent exponential profile likelihood
#'
#' @param x an object of class \code{eprof} returned by \code{\link{gpd.pll}} or \code{\link{gev.pll}}.
#' @param ... further arguments to \code{plot}.
#' @return a graph of the (modified) profile likelihoods
#' @references Brazzale, A. R., Davison, A. C. and Reid, N. (2007). \emph{Applied Asymptotics: Case Studies in Small-Sample Statistics}. Cambridge University Press, Cambridge.
#' @references Severini, T. A. (2000). \emph{Likelihood Methods in Statistics}. Oxford University Press, Oxford.
#' @export
plot.eprof <- function(x, ...) {
# plot the profile log-likelihoods
#old.pars <- par(no.readonly = TRUE)
args <- list(...)
lik <- list()
if (is.null(x$pll) && !is.null(x$r)) {
lik$npll <- -x$r^2/2
} else if (!is.null(x$pll)) {
lik$npll <- x$pll - x$maxpll
} else {
stop("Invalid object provided")
}
if (!is.null(x$tem.pll)) {
lik$tem.npll <- x$tem.pll - x$tem.maxpll
}
if (!is.null(x$empcov.pll)) {
lik$empcov.npll <- x$empcov.pll - x$empcov.maxpll
}
if(is.null(args$ylim)){
ylim <- c(max(-8, min(unlist(lapply(lik, min, na.rm = TRUE)))), 0)
} else{
ylim <- args$ylim
}
tikz <- FALSE
level <- c(0.95, 0.99)
if (!is.null(args$level)) {
level <- args$level[1]
}
if (!is.null(args$tikz)) {
if (isTRUE(args$tikz)) {
tikz <- TRUE
}
}
if (any(!is.null(args$which), !is.null(args$ind), !is.null(args$parm))) {
if (!is.null(args$parm)) {
parm <- match.arg(arg = args$parm, choices = c("profile", "tem", "modif.tem", "modif.empcov",
"r", "rstar"), several.ok = TRUE)
parm[parm %in% "r"] <- "profile"
parm[parm %in% "rstar"] <- "tem"
ind <- which(c("profile", "tem", "modif.tem", "modif.empcov") %in% parm)
} else if (!is.null(args$ind)) {
ind <- args$ind[args$ind %in% 1:4]
parm <- c("profile", "tem", "modif.tem", "modif.empcov")[ind]
} else if (!is.null(args$which)) {
ind <- args$which[args$which %in% 1:4]
parm <- c("profile", "tem", "modif.tem", "modif.empcov")[ind]
}
parm <- unique(parm)
ind <- unique(ind[ind %in% 1:4])
} else {
ind <- 1:4
parm <- c("profile", "tem", "modif.tem", "modif.empcov")
}
if(is.null(args$xlim)){
xlim <- c(min(x$psi), max(x$psi))
} else{
xlim <- args$xlim
}
plot(NULL, type = "n", bty = "l", xlim = xlim, ylim = ylim, xlab = ifelse(!tikz,
expression(psi), "$\\psi$"), ylab = "Profile log likelihood")
abline(h = -qchisq(level, 1)/2, col = "gray")
# Legend
lcols <- NULL
llty <- NULL
llwd <- NULL
llegend <- NULL
if (4 %in% ind && !is.null(x$empcov.mle)) {
abline(v = x$empcov.mle, lwd = 0.5, col = 4, lty = 4)
}
if (3 %in% ind && !is.null(x$tem.mle)) {
abline(v = x$tem.mle, lwd = 0.5, col = 2, lty = 2)
}
if (2 %in% ind && !is.null(x$tem.psimax)) {
abline(v = x$tem.psimax, lwd = 0.5, col = 3)
}
abline(v = x$psi.max, lwd = 0.5)
if (4 %in% ind && !is.null(lik$empcov.npll)) {
lines(x$psi, lik$empcov.npll, lty = 4, col = 4, lwd = 2)
lcols <- c(lcols, 4)
llty <- c(llty, 4)
llwd <- c(llwd, 2)
llegend <- c(llegend, "modif. emp. cov.")
}
if (3 %in% ind && !is.null(lik$tem.npll)) {
lines(x$psi, lik$tem.npll, lty = 2, col = 2, lwd = 2)
lcols <- c(lcols, 2)
llty <- c(llty, 2)
llwd <- c(llwd, 2)
llegend <- c(llegend, "modif. tem.")
}
if (2 %in% ind && !is.null(x$rstar)) {
lines(x$psi, -x$rstar^2/2, lwd = 2, col = 3, lty = 5)
lcols <- c(lcols, 3)
llty <- c(llty, 5)
llwd <- c(llwd, 2)
llegend <- c(llegend, "tem")
}
lcols <- c(lcols, 1)
llty <- c(llty, 1)
llwd <- c(llwd, 2)
llegend <- c(llegend, "profile")
lines(x$psi, lik$npll, lwd = 2)
# add the legend in the top right corner
if(!isTRUE(all.equal(llegend, "profile"))){
legend(x = "topright", legend = rev(llegend), lty = rev(llty), lwd = rev(llwd), col = rev(lcols),
bty = "n", x.intersp = 0.2, seg.len = 0.5, cex = 0.9)
}
#par(old.pars)
}
#' Profile log-likelihood for the generalized extreme value distribution
#'
#' This function calculates the profile likelihood along with two small-sample corrections
#' based on Severini's (1999) empirical covariance and the Fraser and Reid tangent exponential
#' model approximation.
#'
#' @details The two additional \code{mod} available are \code{tem}, the tangent exponential model (TEM) approximation and
#' \code{modif} for the penalized profile likelihood based on \eqn{p^*} approximation proposed by Severini.
#' For the latter, the penalization is based on the TEM or an empirical covariance adjustment term.
#'
#' @param psi parameter vector over which to profile (unidimensional)
#' @param param string indicating the parameter to profile over
#' @param mod string indicating the model, one of \code{profile}, \code{tem} or \code{modif}.See \bold{Details}.
#' @param dat sample vector
#' @param N size of block over which to take maxima. Required only for \code{param} \code{Nmean} and \code{Nquant}.
#' @param p tail probability. Required only for \code{param} \code{quant}.
#' @param q probability level of quantile. Required only for \code{param} \code{Nquant}.
#' @param correction logical indicating whether to use \code{spline.corr} to smooth the tem approximation.
#' @param plot logical; should the profile likelihood be displayed? Default to \code{TRUE}
#' @param ... additional arguments such as output from call to \code{Vfun} if \code{mode='tem'}.
#'
#' @return a list with components
#' \itemize{
#' \item{\code{mle}:} maximum likelihood estimate
#' \item{\code{psi.max}:} maximum profile likelihood estimate
#' \item{\code{param}:} string indicating the parameter to profile over
#' \item{\code{std.error}:} standard error of \code{psi.max}
#' \item{\code{psi}:} vector of parameter \eqn{psi} given in \code{psi}
#' \item{\code{pll}:} values of the profile log likelihood at \code{psi}
#' \item{\code{maxpll}:} value of maximum profile log likelihood
#' }
#'
#'
#' In addition, if \code{mod} includes \code{tem}
#' \itemize{
#' \item{\code{normal}:}{maximum likelihood estimate and standard error of the interest parameter \eqn{psi}}
#' \item{\code{r}:}{values of likelihood root corresponding to \eqn{\psi}}
#' \item{\code{q}:}{vector of likelihood modifications}
#' \item{\code{rstar}:}{modified likelihood root vector}
#' \item{\code{rstar.old}:}{uncorrected modified likelihood root vector}
#' \item{\code{tem.psimax}:}{maximum of the tangent exponential model likelihood}
#' }
#' In addition, if \code{mod} includes \code{modif}
#' \itemize{
#' \item{\code{tem.mle}:} maximum of tangent exponential modified profile log likelihood
#' \item{\code{tem.profll}:} values of the modified profile log likelihood at \code{psi}
#' \item{\code{tem.maxpll}:} value of maximum modified profile log likelihood
#' \item{\code{empcov.mle}:} maximum of Severini's empirical covariance modified profile log likelihood
#' \item{\code{empcov.profll}:} values of the modified profile log likelihood at \code{psi}
#' \item{\code{empcov.maxpll}:} value of maximum modified profile log likelihood
#' }
#'
#' @references Fraser, D. A. S., Reid, N. and Wu, J. (1999), A simple general formula for tail probabilities for frequentist and Bayesian inference. \emph{Biometrika}, \bold{86}(2), 249--264.
#' @references Severini, T. (2000) Likelihood Methods in Statistics. Oxford University Press. ISBN 9780198506508.
#' @references Brazzale, A. R., Davison, A. C. and Reid, N. (2007) Applied asymptotics: case studies in small-sample statistics. Cambridge University Press, Cambridge. ISBN 978-0-521-84703-2
#'
#' @export
#' @examples
#' \dontrun{
#' set.seed(123)
#' dat <- rgev(n = 100, loc = 0, scale = 2, shape = 0.3)
#' gev.pll(psi = seq(0,0.5, length = 50), param = 'shape', dat = dat)
#' gev.pll(psi = seq(-1.5, 1.5, length = 50), param = 'loc', dat = dat)
#' gev.pll(psi = seq(10, 40, length = 50), param = 'quant', dat = dat, p = 0.01)
#' gev.pll(psi = seq(12, 100, length = 50), param = 'Nmean', N = 100, dat = dat)
#' gev.pll(psi = seq(12, 90, length = 50), param = 'Nquant', N = 100, dat = dat, q = 0.5)
#' }
gev.pll <- function(psi, param = c("loc", "scale", "shape", "quant", "Nmean", "Nquant"),
mod = "profile", dat, N = NULL, p = NULL, q = NULL, correction = TRUE, plot = TRUE, ...) {
param <- match.arg(param)
mod <- match.arg(mod, c("profile","tem", "modif"), several.ok = TRUE)
# Parametrization profiling for quant over scale is more numerically stable
if (param == "quant") {
stopifnot(!is.null(p))
q <- 1 - p
N = 1
param <- "Nquant"
}
oldpar <- match.arg(param, choices = c("loc", "scale", "shape", "quant", "Nmean", "Nquant"), several.ok = FALSE)
# Arguments for parametrization of the log likelihood
if (param %in% c("loc", "scale", "shape")) {
args <- c("loc", "scale", "shape")
} else if (param == "quant") {
args <- c(param, "scale", "shape")
} else {
args <- c("loc", param, "shape")
}
# Sanity checks to ensure all arguments are provided
if (is.null(N)) {
if (param %in% c("Nmean", "Nquant")) {
stop("Argument \"N\" missing. Procedure aborted")
} else {
N <- NA
}
}
if (is.null(q)) {
if (param == "Nquant") {
stop("Argument \"q\" missing. Procedure aborted")
} else {
q <- NA
}
}
if (is.null(p)) {
if (param == "quant") {
stop("Argument \"p\" missing. Procedure aborted")
} else {
p <- NA
}
}
xmin <- min(dat)
xmax <- max(dat)
# Find maximum likelihood estimates
mle <- gev.mle(xdat = dat, args = args, q = q, N = N, p = p)
# if(missing(psi) || any(is.null(psi)) || any(is.na(psi))){ psi <- mle[param] } psi <- as.vector(psi)
# Extract the components, notably V for model `tem`. Keep other components for optimization
Vprovided <- FALSE
extra.args <- list(...)
if ("V" %in% names(extra.args)) {
V <- extra.args$V
extra.args$V <- NULL
if (isTRUE(all.equal(dim(V), c(length(dat), 2)))) {
Vprovided <- TRUE
}
}
if (!Vprovided) {
V <- switch(param,
loc = gev.Vfun(par = mle, dat = dat),
scale = gev.Vfun(par = mle, dat = dat),
shape = gev.Vfun(par = mle, dat = dat),
quant = gevr.Vfun(par = mle, dat = dat, p = p),
Nmean = gevN.Vfun(par = mle, dat = dat, N = N, qty = "mean"),
Nquant = gevN.Vfun(par = mle, dat = dat, q = q, N = N, qty = "quantile"))
}
# Obtained constrained maximum likelihood estimates for given value of psi
if (param %in% c("loc", "scale", "shape")) {
# Define observation-wise gradient
gev.score.f <- function(par, dat) {
dat <- as.vector(dat)
mu = par[1]
sigma = par[2]
xi = as.vector(par[3])
if (!isTRUE(all.equal(xi, 0, tolerance = 1e-8))) {
cbind(-(-(mu - dat) * xi/sigma + 1)^(-1/xi - 1)/sigma - xi * (1/xi + 1)/(sigma *
((mu - dat) * xi/sigma - 1)), -(dat - mu) * ((dat - mu) * xi/sigma + 1)^(-1/xi -
1)/sigma^2 + (dat - mu) * xi * (1/xi + 1)/(sigma^2 * ((dat - mu) * xi/sigma +
1)) - 1/sigma, -(mu - dat) * (1/xi + 1)/(sigma * ((mu - dat) * xi/sigma -
1)) - (log1p(-(mu - dat) * xi/sigma)/xi^2 - (mu - dat)/(sigma * ((mu -
dat) * xi/sigma - 1) * xi))/(-(mu - dat) * xi/sigma + 1)^(1/xi) + log1p(-(mu -
dat) * xi/sigma )/xi^2)
} else {
cbind(-exp(mu/sigma - dat/sigma)/sigma + 1/sigma, mu * exp(mu/sigma - dat/sigma)/sigma^2 -
dat * exp(mu/sigma - dat/sigma)/sigma^2 - mu/sigma^2 - 1/sigma + dat/sigma^2,
rep(0, length(dat)))
}
}
ind <- switch(param, loc = 1, scale = 2, shape = 3)
maxll <- gev.ll(mle, dat = dat)
std.error <- sqrt(solve(gev.infomat(par = mle, dat = dat, method = "exp"))[ind, ind])
constr.mle.scale <- function(sigmat, dat = dat) {
x0 = c(median(dat/sigmat), 0.05)
if (is.nan(gev.ll(c(x0[1], sigmat, x0[2]), dat = dat))) {
constr_fit <- try(mev::fit.gev(xdat = dat, fpar = list(scale = sigmat, shape = x0[2])))
if (!inherits(constr_fit, what = "try-error")) {
if (constr_fit$convergence == "successful") {
x0 <- as.vector(c(constr_fit$estimate["loc"], 0.05))
} else {
stop("Could not find starting values for optimization routine")
}
} else {
stop("Could not find starting values for optimization routine")
}
}
# opt <- suppressMessages(nloptr::sbplx(x0 = x0, fn = function(par) {
# -gev.ll(c(par[1], sigmat, par[2]), dat = dat)
# }))
# opt2 <- suppressMessages(nloptr::slsqp(x0 = opt$par, fn = function(par) {
# -gev.ll(c(par[1], sigmat, par[2]), dat = dat)
# }, gr = function(par) {
# -gev.score(c(par[1], sigmat, par[2]), dat = dat)[-2]
# }))
opt <- try(suppressWarnings(suppressMessages(alabama::auglag(par = x0, fn = function(par) {
-gev.ll(c(par[1], sigmat, par[2]), dat = dat)
}, gr = function(par) {
-gev.score(c(par[1], sigmat, par[2]), dat = dat)[-2]
},
hin = function(par) {
ifelse(par[2] <= 0, sigmat + par[2] * (xmax - par[1]), sigmat + par[2] * (xmin -
par[1]))
}, control.outer = list(trace = FALSE, method = "nlminb")))))
if (!inherits(opt, what = "try-error")) {
if (opt$convergence == 0 && !isTRUE(all.equal(opt$value, 1e+10))) {
return(c(opt$par, opt$value))
}
}
opt2 <- try(suppressWarnings(suppressMessages(Rsolnp::solnp(pars = x0, fun = function(par) {
-gev.ll(c(par[1], sigmat, par[2]), dat = dat)
}, control = list(trace = 0)))))
if (!inherits(opt2, what = "try-error")) {
if (opt2$convergence == 0 && !isTRUE(all.equal(tail(opt2$values,1), 1e+10))) {
return(c(opt2$par, tail(opt2$values,1)))
}
}
return(rep(NA, 3))
}
constr.mle.loc <- function(mut, dat = dat) {
opt <- try(suppressMessages(suppressWarnings(
alabama::auglag(par = c(mad(dat, constant = 1), 0.1), fn = function(par) {
val <- -gev.ll(c(mut, par[1:2]), dat = dat)
ifelse(is.infinite(val), 1e+10, val)
}, gr = function(par) {
-gev.score(c(mut, par[1:2]), dat = dat)[-ind]
}, hin = function(par) {
ifelse(par[2] <= 0, par[1] + par[2] * (xmax - mut), par[1] + par[2] * (xmin -
mut))
}, control.outer = list(method = "nlminb", trace = FALSE)))))
if (!inherits(opt, what = "try-error")) {
if (opt$convergence == 0 && !isTRUE(all.equal(opt$value, 1e+10))) {
return(c(opt$par, opt$value))
}
}
opt2 <- try(suppressWarnings(suppressMessages(
Rsolnp::solnp(pars = c(mad(dat, constant = 1), 0.1),
fun = function(par) {
-gev.ll(c(mut, par), dat = dat)
}, control = list(trace = 0)))))
if (!inherits(opt2, what = "try-error")) {
if (opt2$convergence == 0 && !isTRUE(all.equal(tail(opt2$values,1), 1e+10))) {
return(c(opt2$par, tail(opt2$values,1)))
}
}
return(rep(NA, 3))
}
# constr.mle.shape <- function(xit, dat = dat){
# start.scale <- max(1e-2,mad(dat, constant = 1)/median(abs(mev::rgev(10000, shape=xit)-mev::qgev(0.5, shape=xit))))
# start.loc <- median(dat) - start.scale*ifelse(xit == 0, log(log(2)), (log(2)^(-xit)-1)/xit)
# if(any(c(start.scale + xit*(xmax-start.loc) <= 0, start.scale + xit*(xmin-start.loc) <= 0)))
# {
# if(xit < 0){ start.loc <- start.scale/xit + xmax + 1e-3
# } else { start.loc <- start.scale/xit + xmin + 1e-3 } }
# #Subplex - simplex type algorithm, more robust than Nelder-Mead
# opt <- nloptr::sbplx(x0 = c(start.loc, start.scale),
# fn = function(par){-gev.ll(c(par,xit), dat=dat)}, lower= c(-Inf, 0),
# control = list(xtol_rel = 1e-8, maxeval = 2000, ftol_rel = 1e-10))
# #If solution is not within the region
# if(ifelse(xit < 0, opt$par[2] + xit*(xmax-opt$par[1]) <= 0, opt$par[2] + xit*(xmin-opt$par[1]) <= 0))
# {
# opt <- nloptr::auglag(x0 = c(start.loc, start.scale), fn = function(par){
# val <- gev.ll.optim(c(par[1:2], xit), dat = dat);
# ifelse(is.infinite(val) || is.na(val), 1e10, val)},
# gr = function(par){ val <- -gev.score(c(par[1], exp(par[2]), xit), dat = dat)[-ind]},
# hin = function(par){c(ifelse(xit <= 0, exp(par[2]) + xit*(xmax-par[1]), exp(par[2]) + xit*(xmin-par[1])))} ) }
# if(!inherits(opt, what = "try-error")){
# if(all(c(opt$convergence > 0, abs(gev.score(c(opt$par, xit), dat = dat)[1:2]) < 5e-4))){
# return(c(opt$par, opt$value)) } else { #mev::fgev(start = list(loc = opt$par[1], scale =
# opt$par[2]), shape = xit, # x = dat, method = 'BFGS', control=list(reltol=1e-10, abstol =
# 1e-9)) opt2 <- suppressWarnings(nloptr::slsqp(x0 = opt$par, fn = function(par){ val <-
# gev.ll.optim(c(par[1:2], xit), dat = dat); ifelse(is.infinite(val) || is.na(val), 1e10,
# val)}, gr = function(par){ val <- -gev.score(c(par[1], exp(par[2]), xit), dat =
# dat)[-ind]}, hin = function(par){c(ifelse(xit <= 0, exp(par[2]) + xit*(xmax-par[1]),
# exp(par[2]) + xit*(xmin-par[1])))} )) opt2$par[2] <- exp(opt2$par[2])
# if(all(c(opt2$convergence > 0, !isTRUE(all.equal(opt2$value, 1e10)),
# abs(gev.score(c(opt2$par, xit), dat = dat)[1:2]) < 1e-4))){ return(c(opt2$par,
# opt2$value)) } } } return(rep(NA, 3)) }
constr.mle.shape <- function(xit, dat = dat) {
if (abs(xit) < 1e-08) {
xit <- 0
} #because rgev does not handle this case!
start.scale <- mad(dat, constant = 1)/median(abs(mev::rgev(2000, shape = xit) -
mev::qgev(0.5, shape = xit)))
start.loc <- median(dat) - start.scale * ifelse(xit == 0, log(log(2)), (log(2)^(-xit) -
1)/xit)
if (start.scale + xit * (xmax - start.loc) <= 0) {
if (xit < 0) {
start.loc <- start.scale/xit + xmax + 0.001
} else {
start.loc <- start.scale/xit + xmin + 0.001
}
}
opt <- try(suppressMessages(suppressWarnings(
alabama::auglag(par = c(start.loc, start.scale),
fn = function(par) {
val <- -gev.ll(c(par[1:2], xit), dat = dat)
ifelse(is.infinite(val) || is.na(val), 1e+10, val)
}, gr = function(par) {
-gev.score(c(par[1:2], xit), dat = dat)[-ind]
},
hin = function(par) {
ifelse(xit <= 0, par[2] + xit * (xmax - par[1]), par[2] + xit * (xmin - par[1]))
}, control.outer = list(method = "nlminb", trace = FALSE)))))
if (!inherits(opt, what = "try-error")) {
if (opt$convergence == 0 && !isTRUE(all.equal(opt$value, 1e+10))) {
return(c(opt$par, opt$value))
}
}
opt2 <- try(suppressMessages(suppressWarnings(
Rsolnp::solnp(pars = c(start.loc, start.scale),
fun = function(par) {
val <- -gev.ll(c(par[1:2], xit), dat = dat)
ifelse(is.infinite(val) || is.na(val), 1e+10, val)
}, ineqfun = function(par) {
ifelse(xit <= 0, par[2] + xit * (xmax - par[1]), par[2] + xit * (xmin -
par[1]))
}, ineqLB = 0, ineqUB = Inf, control = list(trace = 0)))))
if (!inherits(opt2, what = "try-error")) {
if (isTRUE(opt2$convergence == 0) && !isTRUE(all.equal(tail(opt2$values,1), 1e+10))) {
return(c(opt2$par, tail(opt2$values,1)))
}
}
return(rep(NA, 3))
}
# Missing psi vector
if (missing(psi) || any(is.null(psi)) || any(is.na(psi))) {
if (ind == 2) {
psirangelow <- unique(min(1e-05, seq(-4, -1.5, length = 10) * std.error + mle[param]))
} else {
psirangelow <- seq(-4, -1.5, length = 10) * std.error + mle[param]
}
lowvals <- -t(sapply(psirangelow, function(par) {
switch(param, loc = constr.mle.loc(par, dat = dat), scale = constr.mle.scale(par,
dat = dat), shape = constr.mle.shape(par, dat = dat))
}))[, 3] - maxll
psirangehigh <- seq(1.5, 4, length = 10) * std.error + mle[param]
highvals <- -t(sapply(psirangehigh, function(par) {
switch(param, loc = constr.mle.loc(par, dat = dat), scale = constr.mle.scale(par,
dat = dat), shape = constr.mle.shape(par, dat = dat))
}))[, 3] - maxll
lo <- approx(x = lowvals, y = psirangelow, xout = -4)$y
hi <- approx(x = highvals, y = psirangehigh, xout = -4)$y
lo <- ifelse(is.na(lo), predict(lm(psirangelow ~ lowvals), newdata = data.frame(lowvals = -4))[[1]],
lo)
hi <- ifelse(is.na(hi), predict(lm(psirangehigh ~ highvals), newdata = data.frame(highvals = -4))[[1]],
hi)
psi <- seq(lo, hi, length = 55)
}
if (param == "shape" && min(psi) < -1) {
warning("psi includes shape parameters below -1. These will be automatically removed.")
psi <- psi[psi > -1]
}
# Calculate profile likelihood at psi
lambda <- t(sapply(psi, function(par) {
switch(param, loc = constr.mle.loc(par, dat = dat), scale = constr.mle.scale(par,
dat = dat), shape = constr.mle.shape(par, dat = dat))
}))
pars <- switch(param, loc = cbind(psi, lambda[, 1:2, drop = FALSE]), scale = cbind(lambda[,
1, drop = FALSE], psi, lambda[, 2, drop = FALSE]), shape = cbind(lambda[, 1:2,
drop = FALSE], psi))
# Profile log likelihood values for psi
profll <- -lambda[, 3]
r <- sign(mle[param] - psi) * sqrt(2 * (maxll - profll))
if ("tem" %in% mod) {
# Tangent exponential model approximation of Fraser and Reid to the profile likelihood
phi.mle <- gev.phi(par = mle, dat = dat, V = V)
q2num <- ifelse(ind%%2 == 0, -1, 1) * apply(pars, 1, function(par) {
det(rbind(c(c(phi.mle) - gev.phi(par = par, dat = dat, V = V)), gev.dphi(par = par,
dat = dat, V = V)[-ind, ]))
})
if (isTRUE(any(sign(q2num) * sign(r) < 0, na.rm = TRUE))) {
warning("Correction factor and likelihood root are of opposite sign - check output")
}
logq <- apply(pars, 1, function(par) {
-0.5 * log(det(gev.infomat(par = par, dat = dat, method = "obs")[-ind, -ind]))
}) + log(abs(q2num))
qmlecontrib <- -log(det(gev.dphi(par = mle, dat = dat, V = V))) + 0.5 * log(det(gev.infomat(par = mle,
dat = dat, method = "obs")))
logq <- logq + qmlecontrib
qcor <- sign(q2num) * exp(logq)
rstar <- ifelse(r == 0, 0, r + (logq - log(abs(r)))/r)
###
tem.max.opt <- function(psi, dat = dat) {
lambda <- switch(param, loc = constr.mle.loc(psi, dat = dat), scale = constr.mle.scale(psi,
dat = dat), shape = constr.mle.shape(psi, dat = dat))[1:2]
para <- switch(ind, c(psi, lambda), c(lambda[1], psi, lambda[2]), c(lambda,
psi))
pll <- gev.ll(par = para, dat = dat)
rs <- 2 * (maxll - pll)
logq <- -0.5 * log(det(gev.infomat(par = para, dat = dat, method = "obs")[-ind,
-ind])) + qmlecontrib + log(abs(det(rbind(c(c(phi.mle) - gev.phi(par = para,
dat = dat, V = V)), gev.dphi(par = para, dat = dat, V = V)[-ind, ]))))
abs(rs + logq - log(sqrt(abs(rs))))
# rs is r squared - finding the maximum by multiplying rstar by r
# when rstar vanishes at zero, so does rstar*r
# this ensures that we do not divide by r = zero
}
tem.max <- optim(par = mle[ind], fn = tem.max.opt, method = "Brent", dat = dat,
lower = ifelse(ind == 2, max(1e-05, mle[ind] - std.error), mle[ind] - std.error),
upper = mle[ind] + std.error, control = list(abstol = 1e-10))$par
###
}
# Modified profile likelihood based on p* approximation, two modifications due to Severini
if ("modif" %in% mod) {
tem.objfunc <- function(par, dat = dat) {
0.5 * log(det(gev.infomat(par, dat = dat, method = "obs")[-ind, -ind])) -
log(abs(det(gev.dphi(par = par, dat = dat, V = V[, -ind])[-ind, ])))
}
optim.tem.fn <- function(psi, dat = dat, param = param) {
theta.psi.opt <- switch(param, loc = constr.mle.loc(psi, dat = dat), scale = constr.mle.scale(psi,
dat = dat), shape = constr.mle.shape(psi, dat = dat))
para <- switch(param, loc = c(psi, theta.psi.opt[1:2]), scale = c(theta.psi.opt[1],
psi, theta.psi.opt[2]), shape = c(theta.psi.opt[1:2], psi))
ll <- -theta.psi.opt[3]
ll + tem.objfunc(para, dat = dat)
}
# TEM profile log likelihood values for psi
proflltem <- profll + apply(pars, 1, tem.objfunc, dat = dat)
# Maximum objective function for TEM (line search in neighborhood of the MLE)
tem.mle.opt <- optim(par = mle[ind], fn = optim.tem.fn, method = "Brent", dat = dat,
param = param, lower = ifelse(param == "scale", max(1e-05, mle[ind] - std.error),
mle[ind] - std.error), upper = mle[ind] + std.error, control = list(fnscale = -1))
# Severini empirical covariance function adjustment to profile likelihood Score function -
# observation-wise
# Score at MLE (sums to zero)
score.scale.mle <- gev.score.f(mle, dat)[, -ind] #keep s_lambda
empcov.objfunc <- function(par, dat) {
0.5 * log(det(gev.infomat(par = par, dat = dat, method = "obs")[-ind, -ind])) -
log(abs(sum(score.scale.mle * gev.score.f(par, dat)[, -ind])))
}
profllempcov <- profll + apply(pars, 1, empcov.objfunc, dat = dat)
optim.empcov.fn <- function(psi, param = param, dat = dat) {
theta.psi.opt <- switch(param, loc = constr.mle.loc(psi, dat = dat), scale = constr.mle.scale(psi,
dat = dat), shape = constr.mle.shape(psi, dat = dat))
para <- switch(param, loc = c(psi, theta.psi.opt[1:2]), scale = c(theta.psi.opt[1],
psi, theta.psi.opt[2]), shape = c(theta.psi.opt[1:2], psi))
ll <- -theta.psi.opt[3]
ll + empcov.objfunc(para, dat = dat)
}
empcov.mle.opt <- optim(par = mle[ind], fn = optim.empcov.fn, method = "Brent",
dat = dat, param = param, lower = ifelse(param == "scale", max(1e-05, mle[ind] -
std.error), mle[ind] - std.error), upper = mle[ind] + std.error, control = list(fnscale = -1))
}
# Return levels, quantiles or value-at-risk
} else if (param %in% c("Nquant", "Nmean")) {
qty <- switch(param, Nquant = "quantile", Nmean = "mean")
maxll <- gevN.ll(mle, dat = dat, N = N, q = q, qty = qty)
std.error <- sqrt(solve(gevN.infomat(par = mle, dat = dat, method = "exp", N = N, q = q,
qty = qty))[2, 2])
constr.mle.N <- function(zt, dat = dat) {
st_vals <- c(median(dat), 0.1)
if (isTRUE(as.vector(mle["shape"] > 0))) {
st_vals <- mle[c("loc", "shape")]
}
# COBYLA unfortunately hangs from time to time. so it cannot be used in optimization
# routine without risking hanging despite the algorithm begin more robust and faster for
# this problem...
opt <- try(suppressMessages(suppressWarnings(alabama::auglag(par = st_vals, fn = function(par) {
val <- -gevN.ll(par = c(par[1], zt, par[2]), dat = dat, q = q, qty = qty, N = N)
ifelse(isTRUE(any(is.infinite(val), is.na(val), val <= -maxll)), 1e+10, val)
}, hin = function(par) {
sigma <- switch(qty, quantile = (zt - par[1]) * par[2]/(N^par[2] * (log(1/q))^(-par[2]) -
1), mean = (zt - par[1]) * par[2]/(N^par[2] * gamma(1 - par[2]) - 1))
c(sigma, sigma + par[2] * (xmax - par[1]), sigma + par[2] * (xmin - par[1]))
}, control.outer = list(method = "nlminb", trace = FALSE)))))
#With `alabama` package opt <- try(suppressWarnings( alabama::auglag(par = c(median(dat),
# 0.1), fn = function(par){ val <- -gevN.ll(par = c(par[1], zt, par[2]), dat = dat, q = q,
# qty = qty, N = N); ifelse(is.infinite(val) || is.na(val), 1e10, val)}, gr =
# function(par){-gevN.score(par = c(par[1], zt, par[2]), dat = dat, q = q, qty = qty, N =
# N)[-2]}, hin = function(par){ sigma <- switch(qty, quantile =
# (zt-par[1])*par[2]/(N^par[2]*(log(1/q))^(-par[2])-1), mean =
# (zt-par[1])*par[2]/(N^par[2]*gamma(1-par[2])-1)) c(sigma, sigma + par[2]*(xmax-par[1]),
# sigma + par[2]*(xmin-par[1]))}, control.outer = list (trace = FALSE) )))
if (!inherits(opt, what = "try-error")) {
# if(opt$convergence == 0 && !isTRUE(all.equal(opt$value, 1e10))){
if (isTRUE(opt$convergence == 0) && !isTRUE(all.equal(opt$value, 1e+10))) {
return(c(opt$par, opt$value))
} else{
st_vals <- opt$par
}
}
opt2 <- try(suppressMessages(suppressWarnings(
Rsolnp::solnp(par = st_vals, fun = function(par) {
val <- -gevN.ll(par = c(par[1], zt, par[2]), dat = dat, q = q, qty = qty,
N = N)
ifelse(is.infinite(val) || is.na(val), 1e+10, val)
}, ineqfun = function(par) {
sigma <- switch(qty, quantile = (zt - par[1]) * par[2]/(N^par[2] * (log(1/q))^(-par[2]) -
1), mean = (zt - par[1]) * par[2]/(N^par[2] * gamma(1 - par[2]) - 1))
c(sigma, sigma + par[2] * (xmax - par[1]), sigma + par[2] * (xmin - par[1]))
}, ineqLB = rep(0, 3), ineqUB = rep(Inf, 3),
control = list(trace = 0)))))
if (!inherits(opt2, what = "try-error")) {
if (isTRUE(opt2$convergence == 0) && !isTRUE(all.equal(tail(opt2$values,1), 1e+10))) {
return(c(opt2$par, tail(opt2$values,1)))
}
}
return(rep(NA,3))
}
# Missing psi vector
if (missing(psi) || any(is.null(psi)) || any(is.na(psi))) {
psirangelow <- pmax(1e-05, seq(ifelse(mle[3] < 0, -3.5, -2.5), -0.5, length = 6) *
std.error + mle[2])
lowvals <- -sapply(psirangelow, constr.mle.N, dat = dat)[3, ] - maxll
psirangehigh <- seq(2.5, (1 + mle[3]) * 8, length = 10) * std.error + mle[2]
highvals <- -sapply(psirangehigh, constr.mle.N, dat = dat)[3, ] - maxll
lo <- approx(x = lowvals, y = psirangelow, xout = -4)$y
hi <- approx(x = highvals, y = psirangehigh, xout = -4)$y
lo <- ifelse(is.na(lo), lm(psirangelow ~ lowvals)$coef[2] * -4 + mle[2], lo)
hi <- ifelse(is.na(hi), lm(psirangehigh ~ highvals)$coef[2] * -4 + mle[2], hi)
psi <- c(seq(lo, mle[2], length = 20)[-20], seq(mle[2], hi, length = 55)[-1])
}
lambda <- t(sapply(psi, constr.mle.N, dat = dat))
pars <- cbind(lambda[, 1, drop = FALSE], psi, lambda[, 2, drop = FALSE])
# Profile log likelihood values for psi
profll <- -lambda[, 3]
r <- sign(mle[param] - psi) * sqrt(2 * (maxll - profll))
if ("tem" %in% mod) {
phi.mle <- gevN.phi(par = mle, dat = dat, V = V, N = N, q = q, qty = qty)
q2num <- apply(pars, 1, function(par) {
-det(rbind(c(c(phi.mle) - gevN.phi(par = par, dat = dat, V = V, N = N, q = q,
qty = qty)), gevN.dphi(par = par, dat = dat, V = V, N = N, q = q, qty = qty)[-2,
]))
})
if (isTRUE(any(sign(q2num) * sign(r) < 0, na.rm = TRUE))) {
warning("Correction factor and likelihood root are of opposite sign - check output")
}
logq <- apply(pars, 1, function(par) {
-0.5 * log(det(gevN.infomat(par = par, dat = dat, method = "obs", N = N, q = q,
qty = qty)[-2, -2]))
}) + log(abs(q2num))
qmlecontrib <- -log(det(gevN.dphi(par = mle, dat = dat, V = V, N = N, q = q, qty = qty))) +
0.5 * log(det(gevN.infomat(par = mle, dat = dat, method = "obs", N = N, q = q,
qty = qty)))
logq <- logq + qmlecontrib
qcor <- sign(q2num) * exp(logq)
rstar <- ifelse(r == 0, 0, r + (logq - log(abs(r)))/r)
tem.max.opt <- function(psi, dat = dat) {
lam <- constr.mle.N(psi, dat = dat)
para <- c(lam[1], psi, lam[2])
pll <- -lam[3]
rs <- 2 * (maxll - pll)
logq <- -0.5 * log(det(gevN.infomat(par = para, dat = dat, method = "obs",
N = N, q = q, qty = qty)[-2, -2])) + qmlecontrib + log(abs(det(rbind(c(c(phi.mle) -
gevN.phi(par = para, dat = dat, V = V, q = q, N = N, qty = qty)), gevN.dphi(par = para,
dat = dat, V = V, q = q, N = N, qty = qty)[-2, ]))))
abs(rs + logq - log(sqrt(abs(rs))))
}
tem.max <- optim(par = mle[2], fn = tem.max.opt, method = "Brent", dat = dat, lower = max(1e-05,
mle[2] - std.error), upper = mle[2] + std.error, control = list(abstol = 1e-10))$par
names(mle)[2] <- oldpar
}
if ("modif" %in% mod) {
# Tangent exponential model approximation of Fraser and Reid to the profile likelihood
tem.objfunc.N <- function(par, dat = dat, N = N, qty = qty, q = q) {
0.5 * log(det(gevN.infomat(par = par, dat = dat, method = "obs", N = N, qty = qty,
q = q)[-2, -2])) - log(abs(det(gevN.dphi(par = par, dat = dat, N = N, V = V[,
-2], qty = qty, q = q)[-2, ])))
}
optim.tem.fn.N <- function(psi, dat = dat, q = q, qty = qty, N = N) {
theta.psi.opt <- constr.mle.N(psi, dat = dat)
param <- c(theta.psi.opt[1], psi, theta.psi.opt[2])
ll <- gevN.ll(param, dat = dat, N = N, q = q, qty = qty)
ll + tem.objfunc.N(param, dat = dat, N = N, qty = qty, q = q)
}
# TEM profile log likelihood values for psi
proflltem <- profll + suppressWarnings(apply(pars, 1, tem.objfunc.N, dat = dat,
N = N, qty = qty, q = q))
# Maximum objective function for TEM
tem.mle.opt <- optim(par = mle[2], fn = optim.tem.fn.N, method = "Brent", dat = dat,
q = q, qty = qty, N = N, lower = max(min(dat), mle[2] - std.error), upper = mle[2] +
std.error, control = list(fnscale = -1))
# Severini empirical covariance function adjustment to profile likelihood Score function -
# observation-wise for xi
gevN.score.f <- function(par, dat, N, q = q, qty = qty) {
qty <- match.arg(qty, c("mean", "quantile"))
mu = par[1]
z = par[2]
xi = par[3]
Npxi <- N^xi
log1q <- log(1/q)
logN <- log(N)
if (qty == "quantile") {
# quantiles at prob. q
cbind(((-(Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu - z) + 1)^(-1/xi -
1) * ((Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu - z)^2 + (Npxi * log1q^(-xi) -
1)/(mu - z))/xi + ((Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu - z)^2 +
(Npxi * log1q^(-xi) - 1)/(mu - z)) * (1/xi + 1)/((Npxi * log1q^(-xi) -
1) * (dat - mu)/(mu - z) - 1) - 1/(mu - z)), (-(Npxi * log1q^(-xi) -
1) * (dat - mu) * (-(Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu - z) +
1)^(-1/xi - 1)/((mu - z)^2 * xi) - (Npxi * log1q^(-xi) - 1) * (dat -
mu) * (1/xi + 1)/(((Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu - z) -
1) * (mu - z)^2) + 1/(mu - z)), ((-(Npxi * log1q^(-xi) - 1) * (dat -
mu)/(mu - z) + 1)^(-1/xi) * ((Npxi * log1q^(-xi) * logN - Npxi * log1q^(-xi) *
log(log1q)) * (dat - mu)/(((Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu -
z) - 1) * (mu - z) * xi) - log1p(-(Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu -
z))/xi^2) - (Npxi * log1q^(-xi) * logN - Npxi * log1q^(-xi) *
log(log1q)) * (dat - mu) * (1/xi + 1)/(((Npxi * log1q^(-xi) - 1) *
(dat - mu)/(mu - z) - 1) * (mu - z)) + (Npxi * log1q^(-xi) - 1) * ((Npxi *
log1q^(-xi) * logN - Npxi * log1q^(-xi) * log(log1q)) * (mu -
z) * xi/(Npxi * log1q^(-xi) - 1)^2 - (mu - z)/(Npxi * log1q^(-xi) -
1))/((mu - z) * xi) + log1p(-(Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu -
z))/xi^2))
} else {
# Mean
cbind(((-(Npxi * gamma(-xi + 1) - 1) * (dat - mu)/(mu - z) + 1)^(-1/xi -
1) * ((Npxi * gamma(-xi + 1) - 1) * (dat - mu)/(mu - z)^2 + (Npxi * gamma(-xi +
1) - 1)/(mu - z))/xi + ((Npxi * gamma(-xi + 1) - 1) * (dat - mu)/(mu -
z)^2 + (Npxi * gamma(-xi + 1) - 1)/(mu - z)) * (1/xi + 1)/((Npxi * gamma(-xi +
1) - 1) * (dat - mu)/(mu - z) - 1) - 1/(mu - z)), (-(Npxi * gamma(-xi +
1) - 1) * (dat - mu) * (-(Npxi * gamma(-xi + 1) - 1) * (dat - mu)/(mu -
z) + 1)^(-1/xi - 1)/((mu - z)^2 * xi) - (Npxi * gamma(-xi + 1) - 1) * (dat -
mu) * (1/xi + 1)/(((Npxi * gamma(-xi + 1) - 1) * (dat - mu)/(mu - z) -
1) * (mu - z)^2) + 1/(mu - z)), ((-(Npxi * gamma(-xi + 1) - 1) * (dat -
mu)/(mu - z) + 1)^(-1/xi) * ((Npxi * logN * gamma(-xi + 1) - Npxi * psigamma(-xi +
1) * gamma(-xi + 1)) * (dat - mu)/(((Npxi * gamma(-xi + 1) - 1) * (dat -
mu)/(mu - z) - 1) * (mu - z) * xi) - log1p(-(Npxi * gamma(-xi + 1) - 1) *
(dat - mu)/(mu - z))/xi^2) - (Npxi * logN * gamma(-xi + 1) - Npxi *
psigamma(-xi + 1) * gamma(-xi + 1)) * (dat - mu) * (1/xi + 1)/(((Npxi *
gamma(-xi + 1) - 1) * (dat - mu)/(mu - z) - 1) * (mu - z)) + (Npxi * gamma(-xi +
1) - 1) * ((Npxi * logN * gamma(-xi + 1) - Npxi * psigamma(-xi + 1) *
gamma(-xi + 1)) * (mu - z) * xi/(Npxi * gamma(-xi + 1) - 1)^2 - (mu - z)/(Npxi *
gamma(-xi + 1) - 1))/((mu - z) * xi) + log1p(-(Npxi * gamma(-xi + 1) - 1) *
(dat - mu)/(mu - z))/xi^2))
}
}
# Score at MLE (sums to zero)
score.N.mle <- gevN.score.f(mle, dat, N, qty = qty, q = q)
empcov.objfunc.N <- function(par, dat = dat, q = q, qty = qty, N = N) {
0.5 * log(det(gevN.infomat(par = par, dat = dat, method = "obs", N = N, q = q,
qty = qty)[-2, -2])) - log(abs(sum(score.N.mle * gevN.score.f(par, dat, N = N,
qty = qty, q = q))))
}
profllempcov <- profll + suppressWarnings(apply(pars, 1, empcov.objfunc.N, N = N,
q = q, dat = dat, qty = qty))
optim.empcov.fn.N <- function(psi, dat = dat, q = q, qty = qty, N = N) {
theta.psi.opt <- constr.mle.N(psi, dat = dat)
param <- c(theta.psi.opt[1], psi, theta.psi.opt[2])
ll <- gevN.ll(param, dat = dat, N = N, q = q, qty = qty)
ll + empcov.objfunc.N(param, dat = dat, q = q, qty = qty, N = N)
}
empcov.mle.opt <- optim(par = mle[2], fn = optim.empcov.fn.N, method = "Brent",
dat = dat, qty = qty, q = q, N = N, lower = max(min(dat), mle[2] - std.error),
upper = mle[2] + std.error, control = list(fnscale = -1))
}
}
# Return profile likelihood and quantities of interest (modified likelihoods)
colnames(pars) <- names(mle)
ans <- list(mle = mle, pars = pars, psi.max = as.vector(mle[oldpar]), param = oldpar, std.error = std.error,
psi = psi, pll = profll, maxpll = maxll, r = r)
if ("tem" %in% mod) {
ans$q <- qcor
ans$rstar <- rstar
ans$normal <- c(ans$psi.max, ans$std.error)
if (correction && length(psi) > 10) {
ans <- spline.corr(ans)
}
ans$tem.psimax <- tem.max
}
if ("modif" %in% mod) {
ans$tem.mle <- tem.mle.opt$par
ans$tem.pll <- proflltem
ans$tem.maxpll <- tem.mle.opt$value
ans$empcov.mle <- empcov.mle.opt$par
ans$empcov.pll <- profllempcov
ans$empcov.maxpll <- empcov.mle.opt$value
}
if ("tem" %in% mod) {
class(ans) <- c("eprof", "fr")
} else {
class(ans) <- "eprof"
}
ans$family <- "gev"
if(plot){
plot(ans)
}
return(invisible(ans))
}
#' Profile log-likelihood for the generalized Pareto distribution
#'
#' This function calculates the (modified) profile likelihood based on the \eqn{p^*} formula.
#' There are two small-sample corrections that use a proxy for
#' \eqn{\ell_{\lambda; \hat{\lambda}}}{the sample space derivative of the nuisance},
#' which are based on Severini's (1999) empirical covariance
#' and the Fraser and Reid tangent exponential model approximation.
#' @details The three \code{mod} available are \code{profile} (the default), \code{tem}, the tangent exponential model (TEM) approximation and
#' \code{modif} for the penalized profile likelihood based on \eqn{p^*} approximation proposed by Severini.
#' For the latter, the penalization is based on the TEM or an empirical covariance adjustment term.
#'
#' @param psi parameter vector over which to profile (unidimensional)
#' @param param string indicating the parameter to profile over
#' @param mod string indicating the model. See \bold{Details}.
#' @param mle maximum likelihood estimate in \eqn{(\psi, \xi)} parametrization if \eqn{\psi \neq \xi} and \eqn{(\sigma, \xi)} otherwise (optional).
#' @param dat sample vector of excesses, unless \code{threshold} is provided (in which case user provides original data)
#' @param m number of observations of interest for return levels. Required only for \code{args} values \code{'VaR'} or \code{'ES'}
#' @param N size of block over which to take maxima. Required only for \code{args} \code{Nmean} and \code{Nquant}.
#' @param p tail probability, equivalent to \eqn{1/m}. Required only for \code{args} \code{quant}.
#' @param q level of quantile for N-block maxima. Required only for \code{args} \code{Nquant}.
#' @param correction logical indicating whether to use \code{spline.corr} to smooth the tem approximation.
#' @param threshold numerical threshold above which to fit the generalized Pareto distribution
#' @param plot logical; should the profile likelihood be displayed? Default to \code{TRUE}
#' @param ... additional arguments such as output from call to \code{Vfun} if \code{mode='tem'}.
#'
#' @return a list with components
#' \itemize{
#' \item{\code{mle}:} maximum likelihood estimate
#' \item{\code{psi.max}:} maximum profile likelihood estimate
#' \item{\code{param}:} string indicating the parameter to profile over
#' \item{\code{std.error}:} standard error of \code{psi.max}
#' \item{\code{psi}:} vector of parameter \eqn{psi} given in \code{psi}
#' \item{\code{pll}:} values of the profile log likelihood at \code{psi}
#' \item{\code{maxpll}:} value of maximum profile log likelihood
#' \item{\code{family}:} a string indicating "gpd"
#' \item{\code{threshold}:} value of the threshold, by default zero
#' }
#' In addition, if \code{mod} includes \code{tem}
#' \itemize{
#' \item{\code{normal}:}{maximum likelihood estimate and standard error of the interest parameter \eqn{psi}}
#' \item{\code{r}:}{values of likelihood root corresponding to \eqn{\psi}}
#' \item{\code{q}:}{vector of likelihood modifications}
#' \item{\code{rstar}:}{modified likelihood root vector}
#' \item{\code{rstar.old}:}{uncorrected modified likelihood root vector}
#' \item{\code{tem.psimax}:}{maximum of the tangent exponential model likelihood}
#' }
#' In addition, if \code{mod} includes \code{modif}
#' \itemize{
#' \item{\code{tem.mle}:} maximum of tangent exponential modified profile log likelihood
#' \item{\code{tem.profll}:} values of the modified profile log likelihood at \code{psi}
#' \item{\code{tem.maxpll}:} value of maximum modified profile log likelihood
#' \item{\code{empcov.mle}:} maximum of Severini's empirical covariance modified profile log likelihood
#' \item{\code{empcov.profll}:} values of the modified profile log likelihood at \code{psi}
#' \item{\code{empcov.maxpll}:} value of maximum modified profile log likelihood
#' }
#' @export
#' @examples
#' \dontrun{
#' dat <- rgp(n = 100, scale = 2, shape = 0.3)
#' gpd.pll(psi = seq(-0.5, 1, by=0.01), param = 'shape', dat = dat)
#' gpd.pll(psi = seq(0.1, 5, by=0.1), param = 'scale', dat = dat)
#' gpd.pll(psi = seq(20, 35, by=0.1), param = 'quant', dat = dat, p = 0.01)
#' gpd.pll(psi = seq(20, 80, by=0.1), param = 'ES', dat = dat, m = 100)
#' gpd.pll(psi = seq(15, 100, by=1), param = 'Nmean', N = 100, dat = dat)
#' gpd.pll(psi = seq(15, 90, by=1), param = 'Nquant', N = 100, dat = dat, q = 0.5)
#' }
gpd.pll <- function(psi, param = c("scale", "shape", "quant", "VaR", "ES", "Nmean", "Nquant"),
mod = "profile", mle = NULL, dat, m = NULL, N = NULL, p = NULL, q = NULL, correction = TRUE, threshold = NULL, plot = TRUE,
...) {
param <- match.arg(param)
mod <- match.arg(mod, c("profile", "tem", "modif"), several.ok = TRUE)
#If there is a threshold
if(!is.null(threshold)){
stopifnot(is.numeric(threshold), length(threshold) == 1)
if(min(dat) < threshold){
dat <- dat[dat>threshold] - threshold
} else {
dat <- dat - threshold
}
} else {
threshold <- 0
}
# Arguments for parametrization of the log likelihood
if (param == "shape") {
args <- c("scale", "shape")
} else {
args <- c(param, "shape")
}
# Sanity checks to ensure all arguments are provided
if (is.null(N)) {
if (param %in% c("Nmean", "Nquant")) {
stop("Argument \"N\" missing. Procedure aborted")
} else {
N <- NA
}
}
if (is.null(m)) {
if (param %in% c("VaR", "ES")) {
stop("Argument \"m\" missing. Procedure aborted")
} else {
m <- NA
}
}
if (is.null(p)) {
if (param == "quant") {
stop("Argument \"p\" missing. Procedure aborted")
} else {
p <- NA
}
}
if (is.null(q)) {
if (param == "Nquant") {
stop("Argument \"q\" missing. Procedure aborted")
} else {
q <- NA
}
}
xmin <- min(dat)
xmax <- max(dat)
shiftres <- param %in% c("Nmean", "Nquant", "VaR", "quant")
# If maximum likelihood estimates are not provided, find them
if (is.null(mle) || length(mle) != 2) {
mle <- gpd.mle(xdat = dat, args = args, m = m, N = N, p = p, q = q)
}
# Extract the components, notably V for model `tem`. Keep other components for optimization
Vprovided <- FALSE
extra.args <- list(...)
if ("V" %in% names(extra.args)) {
V <- extra.args$V
extra.args$V <- NULL
if (isTRUE(all.equal(dim(V), c(length(dat), 1)))) {
Vprovided <- TRUE
}
}
if (!Vprovided) {
V <- switch(param, scale = gpd.Vfun(par = mle, dat = dat), shape = gpd.Vfun(par = mle,
dat = dat), quant = gpdr.Vfun(par = mle, dat = dat, m = 1/p), VaR = gpdr.Vfun(par = mle,
dat = dat, m = m), ES = gpde.Vfun(par = mle, dat = dat, m = m), Nmean = gpdN.Vfun(par = mle,
dat = dat, N = N), Nquant = gpdr.Vfun(par = mle, dat = dat, m = 1/(1 - q^(1/N))))
}
# Obtained constrained maximum likelihood estimates for given value of psi
if (param == "scale") {
maxll <- gpd.ll(mle, dat = dat)
std.error <- sqrt(solve(gpd.infomat(par = mle, dat = dat, method = "exp"))[1, 1])
constr.mle.scale <- function(sigmat) {
as.vector(suppressWarnings(optim(par = 0.01,
fn = function(par, scale){
-gpd.ll(par = c(scale, par), dat = dat)},
method = "Brent", lower = max(-1, -sigmat/xmax),
upper = min(10, sigmat/xmin), scale = sigmat)$par))
}
# Missing psi vector
if (missing(psi) || any(is.null(psi)) || any(is.na(psi))) {
psirangelow <- pmax(1e-05, seq(-3, -1.5, length = 6) * std.error + mle[1])
lowvals <- sapply(psirangelow, function(par) {
gpd.ll(c(par, constr.mle.scale(par)), dat = dat)
}) - maxll
psirangehigh <- seq(2.5, 4, length = 6) * std.error + mle[1]
highvals <- sapply(psirangehigh, function(par) {
gpd.ll(c(par, constr.mle.scale(par)), dat = dat)
}) - maxll
lo <- approx(x = lowvals, y = psirangelow, xout = -4)$y
hi <- approx(x = highvals, y = psirangehigh, xout = -4)$y
lo <- ifelse(is.na(lo), lm(psirangelow ~ lowvals)$coef[2] * -4 + mle[1], lo)
hi <- ifelse(is.na(hi), lm(psirangehigh ~ highvals)$coef[2] * -4 + mle[1], hi)
psi <- seq(lo, hi, length = 55)
}
if (any(as.vector(psi) < 0)) {
warning("Negative scale values provided.")
psi <- psi[psi > 0]
if (length(psi) == 0) {
psi <- mle[1]
}
}
pars <- cbind(psi, sapply(psi, constr.mle.scale))
# Profile log likelihood values for psi
profll <- apply(pars, 1, function(par) {
gpd.ll(par = par, dat = dat)
})
r <- sign(mle[param] - psi) * sqrt(2 * (maxll - profll))
if ("tem" %in% mod) {
phi.mle <- gpd.phi(par = mle, dat = dat, V = V)
q2num <- apply(pars, 1, function(par) {
det(rbind(c(c(phi.mle) - gpd.phi(par = par, dat = dat, V = V)), gpd.dphi(par = par,
dat = dat, V = V)[-1, ]))
})
if (isTRUE(any(sign(q2num) * sign(r) < 0, na.rm = TRUE))) {
warning("Correction factor and likelihood root are of opposite sign - check output")
}
logq <- apply(pars, 1, function(par) {
-0.5 * log(gpd.infomat(par = par, dat = dat, method = "obs")[-1, -1])
}) + log(abs(q2num))
qmlecontrib <- -log(det(gpd.dphi(par = mle, dat = dat, V = V))) + 0.5 * log(det(gpd.infomat(par = mle,
dat = dat, method = "obs")))
logq <- logq + qmlecontrib
qcor <- sign(q2num) * exp(logq)
rstar <- ifelse(r == 0, 0, r + (logq - log(abs(r)))/r)
tem.max.opt <- function(psi, dat = dat) {
para <- c(psi, constr.mle.scale(psi))
pll <- gpd.ll(par = para, dat = dat)
rs <- 2 * (maxll - pll)
logq <- -0.5 * log(gpd.infomat(par = para, dat = dat, method = "obs")[-1, -1]) +
qmlecontrib + log(abs(det(rbind(c(c(phi.mle) - gpd.phi(par = para, dat = dat,
V = V)), gpd.dphi(par = para, dat = dat, V = V)[-1, ]))))
rs + logq - log(sqrt(abs(rs)))
}
tem.max <- optim(par = mle[1], fn = tem.max.opt, method = "Brent", dat = dat, lower = max(1e-05,
mle[1] - std.error), upper = mle[1] + std.error, control = list(abstol = 1e-10))$par
}
if ("modif" %in% mod) {
# Tangent exponential model approximation of Fraser and Reid to the profile likelihood
tem.objfunc.scale <- function(par) {
0.5 * log(gpd.infomat(par = par, dat = dat, method = "obs")[2, 2]) -
log(abs(gpd.dphi(par = par, dat = dat, V = V[, 2, drop = FALSE])[2, 1]))
}
optim.tem.fn.scale <- function(psi) {
theta.psi.opt <- constr.mle.scale(psi)
param <- c(psi, theta.psi.opt) #ll <- -theta.psi.opt$nllh
ll <- gpd.ll(param, dat = dat)
ll + tem.objfunc.scale(param)
}
# TEM profile log likelihood values for psi
proflltem <- profll + apply(pars, 1, tem.objfunc.scale)
# Maximum objective function for TEM (line search in neighborhood of the MLE)
tem.mle.opt <- optim(par = mle[1], fn = optim.tem.fn.scale, method = "Brent", lower = max(1e-05,
mle[1] - std.error), upper = mle[1] + std.error, control = list(fnscale = -1))
tem.mle <- c(tem.mle.opt$par, constr.mle.scale(tem.mle.opt$par))
# Severini empirical covariance function adjustment to profile likelihood Score function -
# observation-wise
gpd.score.f <- function(par, dat) {
sigma = par[1]
xi = par[2]
if (!isTRUE(all.equal(0, xi))) {
cbind(dat * (xi + 1)/(sigma^2 * (dat * xi/sigma + 1)) - 1/sigma, -dat * (1/xi +
1)/(sigma * (dat * xi/sigma + 1)) + log(pmax(dat * xi/sigma + 1, 0))/xi^2)
} else {
cbind((dat - sigma)/sigma^2, 1/2 * (dat - 2 * sigma) * dat/sigma^2)
}
}
# Score at MLE (sums to zero)
score.scale.mle <- gpd.score.f(mle, dat)[, 2] #keep s_lambda
empcov.objfunc.scale <- function(par) {
0.5 * log(gpd.infomat(par = par, dat = dat, method = "obs")[2, 2]) -
log(abs(sum(score.scale.mle * gpd.score.f(par, dat)[, 2])))
}
profllempcov <- profll + apply(pars, 1, empcov.objfunc.scale)
optim.empcov.fn.scale <- function(psi) {
theta.psi.opt <- constr.mle.scale(psi)
param <- c(psi, theta.psi.opt)
ll <- gpd.ll(param, dat = dat)
ll + empcov.objfunc.scale(param)
}
empcov.mle.opt <- optim(par = mle[1], fn = optim.empcov.fn.scale, method = "Brent",
lower = max(1e-05, mle[1] - std.error), upper = mle[1] + std.error, control = list(fnscale = -1))
empcov.mle <- c(empcov.mle.opt$par, constr.mle.scale(empcov.mle.opt$par))
}
# Shape parameter
} else if (param == "shape") {
maxll <- gpd.ll(mle, dat = dat)
std.error <- sqrt(solve(gpd.infomat(par = mle, dat = dat, method = "exp"))[2, 2])
constr.mle.shape <- function(xit) {
as.vector(suppressWarnings(optim(par = 2 * abs(xit) * xmax, fn = function(par, shape){-gpd.ll(par = c(par, shape), dat = dat)},
method = "Brent", shape = xit, lower = ifelse(xit < 0, abs(xit) * xmax + 1e-05, 1e-05), upper = 1e+10)$par))
}
# Missing psi vector
if (missing(psi) || any(is.null(psi)) || any(is.na(psi))) {
psirangelow <- seq(ifelse(mle[2] < 0, -7, -5), -1.5, length = 10) * std.error + mle[2]
psirangelow <- psirangelow[psirangelow > -1]
lowvals <- sapply(psirangelow, function(par) {
gpd.ll(c(constr.mle.shape(par), par), dat = dat)
}) - maxll
psirangehigh <- seq(1.5, 10, length = 10) * std.error + mle[2]
highvals <- sapply(psirangehigh, function(par) {
gpd.ll(c(constr.mle.shape(par), par), dat = dat)
}) - maxll
lo <- approx(x = lowvals, y = psirangelow, xout = -4)$y
hi <- approx(x = highvals, y = psirangehigh, xout = -4)$y
lo <- ifelse(is.na(lo), lm(psirangelow ~ lowvals)$coef[2] * -4 + mle[2], lo)
hi <- ifelse(is.na(hi), lm(psirangehigh ~ highvals)$coef[2] * -4 + mle[2], hi)
psi <- seq(lo, hi, length = 55)
psi <- psi[psi>-1]
}
pars <- cbind(sapply(psi, constr.mle.shape), psi)
# Profile log likelihood values for psi
profll <- apply(pars, 1, function(par) {
gpd.ll(par = par, dat = dat)
})
r <- sign(mle[param] - psi) * sqrt(2 * (maxll - profll))
if ("tem" %in% mod) {
phi.mle <- gpd.phi(par = mle, dat = dat, V = V)
q2num <- apply(pars, 1, function(par) {
det(rbind(gpd.dphi(par = par, dat = dat, V = V)[-2, ], c(c(phi.mle) - gpd.phi(par = par,
dat = dat, V = V))))
})
if (isTRUE(any(sign(q2num) * sign(r) < 0, na.rm = TRUE))) {
warning("Correction factor and likelihood root are of opposite sign - check output")
}
logq <- apply(pars, 1, function(par) {
-0.5 * log(gpd.infomat(par = par, dat = dat, method = "obs")[-2, -2])
}) + log(abs(q2num))
qmlecontrib <- -log(det(gpd.dphi(par = mle, dat = dat, V = V))) + 0.5 * log(det(gpd.infomat(par = mle,
dat = dat, method = "obs")))
logq <- logq + qmlecontrib
qcor <- sign(q2num) * exp(logq)
rstar <- ifelse(r == 0, 0, r + (logq - log(abs(r)))/r)
# Maximum of TEM likelihood - indirect estimation via rstar
tem.max.opt <- function(psi, dat = dat) {
para <- c(constr.mle.shape(psi), psi)
pll <- gpd.ll(par = para, dat = dat)
rs <- 2 * (maxll - pll)
logq <- -0.5 * log(gpd.infomat(par = para, dat = dat, method = "obs")[-2, -2]) +
qmlecontrib + log(abs(det(rbind(c(c(phi.mle) - gpd.phi(par = para, dat = dat,
V = V)), gpd.dphi(par = para, dat = dat, V = V)[-2, ]))))
rs + logq - log(sqrt(abs(rs)))
}
tem.max <- optim(par = mle[2], fn = tem.max.opt, method = "Brent", dat = dat, lower = mle[2] -
std.error, upper = mle[2] + std.error, control = list(abstol = 1e-10))$par
}
if ("modif" %in% mod) {
# Tangent exponential model approximation of Fraser and Reid to the profile likelihood
tem.objfunc.shape <- function(par) {
0.5 * log(gpd.infomat(par = par, dat = dat, method = "obs")[1, 1]) -
log(abs(gpd.dphi(par = par, dat = dat, V = V[, 1, drop = FALSE])[1, 1]))
}
optim.tem.fn.shape <- function(psi) {
theta.psi.opt <- constr.mle.shape(psi)
param <- c(theta.psi.opt, psi)
ll <- gpd.ll(param, dat = dat)
ll + tem.objfunc.shape(param)
}
# TEM profile log likelihood values for psi
proflltem <- profll + apply(pars, 1, tem.objfunc.shape)
# Maximum objective function for TEM (line search in neighborhood of the MLE)
tem.mle.opt <- optim(par = mle[2], fn = optim.tem.fn.shape, method = "Brent", lower = mle[2] -
std.error, upper = mle[2] + std.error, control = list(fnscale = -1))
tem.mle <- c(constr.mle.shape(tem.mle.opt$par), tem.mle.opt$par)
# Severini empirical covariance function adjustment to profile likelihood Score function -
# observation-wise
gpd.score.f <- function(par, dat) {
sigma = par[1]
xi = par[2]
if (!isTRUE(all.equal(0, xi))) {
cbind(dat * (xi + 1)/(sigma^2 * (dat * xi/sigma + 1)) - 1/sigma, -dat * (1/xi +
1)/(sigma * (dat * xi/sigma + 1)) + log(pmax(dat * xi/sigma + 1, 0))/xi^2)
} else {
cbind((dat - sigma)/sigma^2, 1/2 * (dat - 2 * sigma) * dat/sigma^2)
}
}
# Score at MLE (sums to zero)
score.shape.mle <- gpd.score.f(mle, dat)[, 1] #keep s_lambda
empcov.objfunc.shape <- function(par) {
0.5 * log(gpd.infomat(par = par, dat = dat, method = "obs")[1, 1]) -
log(abs(sum(score.shape.mle * gpd.score.f(par, dat)[, 1])))
}
profllempcov <- profll + apply(pars, 1, empcov.objfunc.shape)
optim.empcov.fn.shape <- function(psi) {
theta.psi.opt <- constr.mle.shape(psi)
param <- c(theta.psi.opt, psi)
ll <- gpd.ll(param, dat = dat)
ll + empcov.objfunc.shape(param)
}
empcov.mle.opt <- optim(par = mle[2], fn = optim.empcov.fn.shape, method = "Brent",
lower = mle[2] - std.error, upper = mle[2] + std.error, control = list(fnscale = -1))
empcov.mle <- c(constr.mle.shape(empcov.mle.opt$par), empcov.mle.opt$par)
}
# Return levels, quantiles or value-at-risk
} else if (param %in% c("quant", "VaR", "Nquant")) {
if (param == "quant") {
m <- 1/p
}
if (param == "Nquant") {
m <- 1/(1 - q^(1/N))
}
maxll <- gpdr.ll(mle, dat = dat, m = m)
std.error <- sqrt(solve(gpdr.infomat(par = mle, dat = dat, method = "exp", m = m))[1,
1])
constr.mle.quant <- function(quant) {
suppressMessages(Rsolnp::solnp(par = 0.01, function(lambda, psi, m) {
-gpdr.ll(par = c(psi, lambda), dat = dat, m = m)
}, psi = quant, m = m, lower = -1, upper = 5)$par)
}
# Missing psi vector
if (missing(psi) || any(is.null(psi)) || any(is.na(psi))) {
psirangelow <- unique(pmax(mean(dat), seq(-3, -0.5, length = 12) * std.error +
mle[1]))
lowvals <- sapply(psirangelow, function(par) {
gpdr.ll(c(par, constr.mle.quant(par)), m = m, dat = dat)
}) - maxll
psirangehigh <- seq(2, 10, length = 12) * std.error + mle[1]
highvals <- sapply(psirangehigh, function(par) {
gpdr.ll(c(par, constr.mle.quant(par)), m = m, dat = dat)
}) - maxll
lo <- approx(x = lowvals, y = psirangelow, xout = -4)$y
hi <- approx(x = highvals, y = psirangehigh, xout = -4)$y
lo <- ifelse(is.na(lo), lm(psirangelow ~ lowvals)$coef[2] * -4 + mle[1], lo)
hi <- ifelse(is.na(hi), lm(psirangehigh ~ highvals)$coef[2] * -4 + mle[1], hi)
psi <- seq(lo, hi, length = 55)
} else{
psi <- psi - threshold
if(any(psi < 0)){
stop("Invalid psi sequence: rescaled psi for quantiles must be positive")
}
}
pars <- cbind(psi, sapply(psi, constr.mle.quant))
# Profile log likelihood values for psi
profll <- apply(pars, 1, function(par) {
gpdr.ll(par = par, dat = dat, m = m)
})
r <- sign(mle[param] - psi) * sqrt(2 * (maxll - profll))
if ("tem" %in% mod) {
phi.mle <- gpdr.phi(par = mle, dat = dat, m = m, V = V)
q2num <- apply(pars, 1, function(par) {
det(rbind(c(c(phi.mle) - gpdr.phi(par = par, dat = dat, V = V, m = m)), gpdr.dphi(par = par,
dat = dat, V = V, m = m)[-1, ]))
})
if (isTRUE(any(sign(q2num) * sign(r) < 0, na.rm = TRUE))) {
warning("Correction factor and likelihood root are of opposite sign - check output")
}
logq <- apply(pars, 1, function(par) {
-0.5 * log(gpdr.infomat(par = par, dat = dat, method = "obs", m = m)[-1, -1])
}) + log(abs(q2num))
qmlecontrib <- -log(det(gpdr.dphi(par = mle, dat = dat, V = V, m = m))) + 0.5 *
log(det(gpdr.infomat(par = mle, dat = dat, method = "obs", m = m)))
logq <- logq + qmlecontrib
qcor <- sign(q2num) * exp(logq)
rstar <- ifelse(r == 0, 0, r + (logq - log(abs(r)))/r)
tem.max.opt <- function(psi, dat = dat) {
para <- c(psi, constr.mle.quant(psi))
pll <- gpdr.ll(par = para, dat = dat, m = m)
rs <- 2 * (maxll - pll)
logq <- -0.5 * log(gpdr.infomat(par = para, dat = dat, method = "obs", m = m)[-1,
-1]) + qmlecontrib + log(abs(det(rbind(c(c(phi.mle) - gpdr.phi(par = para,
dat = dat, V = V, m = m)), gpdr.dphi(par = para, dat = dat, V = V, m = m)[-1,
]))))
rs + logq - log(sqrt(abs(rs)))
}
tem.max <- optim(par = mle[1], fn = tem.max.opt, method = "Brent", dat = dat, lower = max(1e-05,
mle[1] - std.error), upper = mle[1] + std.error, control = list(abstol = 1e-10))$par
}
if ("modif" %in% mod) {
# Tangent exponential model approximation of Fraser and Reid to the profile likelihood
tem.objfunc.quant <- function(par) {
0.5 * log(gpdr.infomat(par = par, dat = dat, method = "obs", m = m)[2, 2]) -
log(abs(gpdr.dphi(par = par, dat = dat, m = m, V = V[, 2, drop = FALSE])[2, 1]))
}
optim.tem.fn.quant <- function(psi) {
theta.psi.opt <- constr.mle.quant(psi)
param <- c(psi, theta.psi.opt)
ll <- gpdr.ll(param, dat = dat, m = m)
ll + tem.objfunc.quant(param)
}
# TEM profile log likelihood values for psi
proflltem <- profll + suppressWarnings(apply(pars, 1, tem.objfunc.quant))
# Maximum objective function for TEM
tem.mle.opt <- optim(par = mle[1], fn = optim.tem.fn.quant, method = "Brent", lower = max(1e-05,
mle[1] - std.error), upper = mle[1] + std.error, control = list(fnscale = -1))
tem.mle <- c(tem.mle.opt$par, constr.mle.quant(tem.mle.opt$par))
# Severini empirical covariance function adjustment to profile likelihood Score function -
# observation-wise for xi
gpdr.score.f <- function(par, dat, m) {
xi = par[2]
r = par[1]
-dat * m^xi * (1/xi + 1) * log(m)/((dat * (m^xi - 1)/r + 1) * r) + (m^xi *
r * xi * log(m)/(m^xi - 1)^2 - r/(m^xi - 1)) * (m^xi - 1)/(r * xi) + log(dat *
(m^xi - 1)/r + 1)/xi^2
}
# Score at MLE (sums to zero)
score.quant.mle <- gpdr.score.f(mle, dat, m)
empcov.objfunc.quant <- function(par) {
0.5 * log(gpdr.infomat(par = par, dat = dat, method = "obs", m = m)[2, 2]) -
log(abs(sum(score.quant.mle * gpdr.score.f(par, dat, m = m))))
}
profllempcov <- profll + suppressWarnings(apply(pars, 1, empcov.objfunc.quant))
optim.empcov.fn.quant <- function(psi) {
theta.psi.opt <- constr.mle.quant(psi)
param <- c(psi, theta.psi.opt)
ll <- gpdr.ll(param, dat = dat, m = m)
ll + empcov.objfunc.quant(param)
}
empcov.mle.opt <- optim(par = mle[1], fn = optim.empcov.fn.quant, method = "Brent",
lower = max(1e-05, mle[1] - std.error), upper = mle[1] + std.error, control = list(fnscale = -1))
empcov.mle <- c(empcov.mle.opt$par, constr.mle.quant(empcov.mle.opt$par))
}
} else if (param == "ES") {
maxll <- gpde.ll(mle, dat = dat, m = m)
std.error <- sqrt(solve(gpde.infomat(par = mle, dat = dat, method = "exp", m = m))[1,
1])
constr.mle.es <- function(psif) {
# nloptr::auglag(x0 = 0.1, fn = function(x){-gpde.ll(par = c(psif, x), dat = dat, m = m)},
# hin = function(x){ c(psif*(1-x)*((m^x-1)/x+1)^(-1),
# 1+psif*(1-x)*((m^x-1)/x+1)^(-1)/x*xmin, 1+psif*(1-x)*((m^x-1)/x+1)^(-1)/x*xmax)}, lower =
# -1+1e-10, upper = 1-1e-5)$par}
optim(par = 0.1, fn = function(x) {
-gpde.ll(par = c(psif, x), dat = dat, m = m)
}, method = "Brent", lower = -1 + 1e-10, upper = 1 - 1e-05)$par
}
# Missing psi vector
if (missing(psi) || any(is.null(psi)) || any(is.na(psi))) {
psirangelow <- unique(pmax(mean(dat), seq(-3, -0.1, length = 15) * std.error +
mle[1]))
lowvals <- sapply(psirangelow, function(par) {
gpde.ll(c(par, constr.mle.es(par)), m = m, dat = dat)
}) - maxll
psirangehigh <- seq(2.5, 30, length = 12) * std.error + mle[1]
highvals <- sapply(psirangehigh, function(par) {
gpde.ll(c(par, constr.mle.es(par)), m = m, dat = dat)
}) - maxll
lo <- approx(x = lowvals, y = psirangelow, xout = -4)$y
hi <- approx(x = highvals, y = psirangehigh, xout = -4)$y
lo <- ifelse(is.na(lo), lm(psirangelow ~ lowvals)$coef[2] * -4 + mle[1], lo)
hi <- ifelse(is.na(hi), lm(psirangehigh ~ highvals)$coef[2] * -4.5 + mle[1], hi)
psi <- c(seq(lo, mle[1], length = 15)[-15], seq(mle[1], min(mle[1] + 2 * std.error,
hi), length = 25)[-1])
if (mle[1] + 2 * std.error < hi) {
psi <- c(psi, seq(mle[1] + 2 * std.error, hi, length = 20))
}
}
# Do not remove threshold for expected shortfall
pars <- cbind(psi, sapply(psi, constr.mle.es))
# Profile log likelihood values for psi
profll <- apply(pars, 1, function(par) {
gpde.ll(par = par, dat = dat, m = m)
})
profll[profll == 1e+10] <- NA
r <- sign(mle[param] - psi) * sqrt(2 * (maxll - profll))
if ("tem" %in% mod) {
phi.mle <- gpde.phi(par = mle, dat = dat, m = m, V = V)
q2num <- apply(pars, 1, function(par) {
det(rbind(c(c(phi.mle) - gpde.phi(par = par, dat = dat, V = V, m = m)), gpde.dphi(par = par,
dat = dat, V = V, m = m)[-1, ]))
})
if (isTRUE(any(sign(q2num) * sign(r) < 0, na.rm = TRUE))) {
warning("Correction factor and likelihood root are of opposite sign - check output")
}
logq <- apply(pars, 1, function(par) {
-0.5 * log(gpde.infomat(par = par, dat = dat, method = "obs", m = m)[-1, -1])
}) + log(abs(q2num))
qmlecontrib <- -log(det(gpde.dphi(par = mle, dat = dat, V = V, m = m))) + 0.5 *
log(det(gpde.infomat(par = mle, dat = dat, method = "obs", m = m)))
logq <- logq + qmlecontrib
qcor <- sign(q2num) * exp(logq)
rstar <- ifelse(r == 0, 0, r + (logq - log(abs(r)))/r)
tem.max.opt <- function(psi, dat = dat) {
para <- c(psi, constr.mle.es(psi))
pll <- gpde.ll(par = para, dat = dat, m = m)
rs <- 2 * (maxll - pll)
logq <- -0.5 * log(gpde.infomat(par = para, dat = dat, method = "obs", m = m)[-1,
-1]) + qmlecontrib + log(abs(det(rbind(c(c(phi.mle) - gpde.phi(par = para,
dat = dat, V = V, m = m)), gpde.dphi(par = para, dat = dat, V = V, m = m)[-1,
]))))
rs + logq - log(sqrt(abs(rs)))
}
tem.max <- optim(par = mle[1], fn = tem.max.opt, method = "Brent", dat = dat, lower = max(1e-05,
mle[1] - std.error), upper = mle[1] + std.error, control = list(abstol = 1e-10))$par
}
if ("modif" %in% mod) {
# Tangent exponential model approximation of Fraser and Reid to the profile likelihood
tem.objfunc.es <- function(par) {
0.5 * log(gpde.infomat(par = par, dat = dat, method = "obs", m = m)[2, 2]) -
log(abs(gpde.dphi(par = par, dat = dat, m = m, V = V[, 2, drop = FALSE])[2, 1]))
}
optim.tem.fn.es <- function(psi) {
theta.psi.opt <- constr.mle.es(psi)
param <- c(psi, theta.psi.opt)
ll <- gpde.ll(param, dat = dat, m = m)
ll + tem.objfunc.es(param)
}
# TEM profile log likelihood values for psi
proflltem <- profll + suppressWarnings(apply(pars, 1, tem.objfunc.es))
# Maximum objective function for TEM
tem.mle.opt <- optim(par = mle[1], fn = optim.tem.fn.es, method = "Brent", lower = max(quantile(dat,
1 - 1/m), mle[1] - std.error), upper = mle[1] + std.error, control = list(fnscale = -1))
tem.mle <- c(tem.mle.opt$par, constr.mle.es(tem.mle.opt$par))
# Severini empirical covariance function adjustment to profile likelihood Score function -
# observation-wise for xi
gpde.score.f <- function(par, dat, m) {
xi = par[2]
es = par[1]
-((m^xi * log(m) + 1) * dat/(es * (xi - 1)) - dat * (m^xi + xi - 1)/(es * (xi -
1)^2)) * (1/xi + 1)/(dat * (m^xi + xi - 1)/(es * (xi - 1)) - 1) + (m^xi +
xi - 1) * ((m^xi * log(m) + 1) * (xi - 1) * xi/(m^xi + xi - 1)^2 - (xi -
1)/(m^xi + xi - 1) - xi/(m^xi + xi - 1))/((xi - 1) * xi) + log(pmax(0, -dat *
(m^xi + xi - 1)/(es * (xi - 1)) + 1))/xi^2
}
# Score at MLE (sums to zero)
score.es.mle <- gpde.score.f(mle, dat, m)
empcov.objfunc.es <- function(par) {
0.5 * log(gpde.infomat(par = par, dat = dat, method = "obs", m = m)[2, 2]) -
log(abs(sum(score.es.mle * gpde.score.f(par, dat, m = m))))
}
profllempcov <- profll + suppressWarnings(apply(pars, 1, empcov.objfunc.es))
optim.empcov.fn.es <- function(psi) {
theta.psi.opt <- constr.mle.es(psi)
param <- c(psi, theta.psi.opt)
ll <- gpde.ll(param, dat = dat, m = m)
ll + empcov.objfunc.es(param)
}
empcov.mle.opt <- optim(par = mle[1], fn = optim.empcov.fn.es, method = "Brent",
lower = max(quantile(dat, 1 - 1/m), mle[1] - std.error), upper = mle[1] + std.error,
control = list(fnscale = -1))
empcov.mle <- c(empcov.mle.opt$par, constr.mle.es(empcov.mle.opt$par))
}
} else if (param == "Nmean") {
maxll <- gpdN.ll(mle, dat = dat, N = N)
std.error <- sqrt(solve(gpdN.infomat(par = mle, dat = dat, method = "exp", N = N))[1])
constr.mle.Nmean <- function(Nmeant) {
suppressWarnings(alabama::auglag(par = 0.01, function(lambda, psi, N) {
-gpdN.ll(par = c(psi, lambda), dat = dat, N = N)
}, gr = function(lambda, psi, N) {
-gpdN.score(par = c(psi, lambda), dat = dat, N = N)[2]
},
hin = function(lambda, psi, N){
sigma = ifelse(abs(lambda > 1e-8),
psi * lambda/(exp(lgamma(N + 1) + lgamma(-lambda + 1) - lgamma(N - lambda + 1)) - 1),
psi / (0.57721566490153231044 + psigamma(N + 1)))
if(lambda >= 0){c(1e-8, sigma, 1-lambda, lambda + 1)} else{ c(- sigma/lambda - xmax, sigma, 1 - lambda, lambda + 1)}},
psi = Nmeant, N = N, control.outer = list(trace = FALSE))$par)
}
# Missing psi vector
if (missing(psi) || any(is.null(psi)) || any(is.na(psi))) {
#compute profile log-lik on a grid left and right of the MLE
psirangelow <- unique(pmax(mean(dat), seq(-3, -0.25, length = 20) * std.error +
mle[1]))
lowvals <- sapply(psirangelow, function(par) {
gpdN.ll(c(par, constr.mle.Nmean(par)), dat = dat, N = N)
}) - maxll
psirangehigh <- seq(2.5, 50, length = 20) * std.error + mle[1]
highvals <- sapply(psirangehigh, function(par) {
gpdN.ll(c(par, constr.mle.Nmean(par)), dat = dat, N = N)
}) - maxll
#Try to do linear interpolation - only works if value inside the interval lowvals or highvals
lo <- approx(x = lowvals, y = psirangelow, xout = -4)$y
#Else linear interpolation with linear model fitted to lower values
lo <- ifelse(is.na(lo), spline(x = lowvals, y = psirangelow, xout = -4)$y, lo)
psirangelow <- seq(lo, mle[1], length = 20)
lowvals <- sapply(psirangelow, function(par) {
gpdN.ll(c(par, constr.mle.Nmean(par)), dat = dat, N = N)
}) - maxll
#hi <- approx(x = highvals, y = psirangehigh, xout = -4)$y
#For upper, use spline approx
hi <- spline(x = highvals, y = psirangehigh, xout = -4)$y
#Recompute the range
psirangehigh <- seq(psirangehigh[1], hi, length = 30)
highvals <- sapply(psirangehigh, function(par) {
gpdN.ll(c(par, constr.mle.Nmean(par)), dat = dat, N = N)
}) - maxll
#Remove NAs, inf, etc.
highvals <- highvals[is.finite(highvals)]
psirangehigh <- psirangehigh[1:length(highvals)]
#If could not interpolate, use a simple linear model to predict lower value
#hi <- ifelse(is.na(hi), spline(x = highvals, y = psirangehigh, xout = -4)$y, hi)
psi <- as.vector(c(spline(x=c(lowvals, 0), y = c(psirangelow, mle[1]), xout = seq(-4, -0.1, length = 15))$y, mle[1],
spline(x=c(0, highvals), y = c(mle[1], psirangehigh), xout = seq(-0.1, highvals[length(highvals)], length = 20))$y))
} else{
psi <- psi - threshold
}
if (any(as.vector(psi) < 0)) {
warning("Negative Nmean values provided.")
psi <- psi[psi > 0]
if (length(psi) == 0) {
psi <- mle[1]
}
}
pars <- cbind(psi, sapply(psi, constr.mle.Nmean))
# Profile log likelihood values for psi
profll <- apply(pars, 1, function(par) {
gpdN.ll(par = par, dat = dat, N = N)
})
r <- sign(mle[param] - psi) * sqrt(2 * (maxll - profll))
if ("tem" %in% mod) {
phi.mle <- gpdN.phi(par = mle, dat = dat, N = N, V = V)
q2num <- apply(pars, 1, function(par) {
det(rbind(c(c(phi.mle) - gpdN.phi(par = par, dat = dat, V = V, N = N)), gpdN.dphi(par = par,
dat = dat, V = V, N = N)[-1, ]))
})
if (isTRUE(any(sign(q2num) * sign(r) < 0, na.rm = TRUE))) {
warning("Correction factor and likelihood root are of opposite sign - check output")
}
logq <- apply(pars, 1, function(par) {
-0.5 * log(gpdN.infomat(par = par, dat = dat, method = "obs", N = N)[-1, -1])
}) + log(abs(q2num))
qmlecontrib <- -log(det(gpdN.dphi(par = mle, dat = dat, V = V, N = N))) + 0.5 *
log(det(gpdN.infomat(par = mle, dat = dat, method = "obs", N = N)))
logq <- logq + qmlecontrib
qcor <- sign(q2num) * exp(logq)
rstar <- ifelse(r == 0, 0, r + (logq - log(abs(r)))/r)
tem.max.opt <- function(psi, dat = dat) {
para <- c(psi, constr.mle.Nmean(psi))
pll <- gpdN.ll(par = para, dat = dat, N = N)
rs <- 2 * (maxll - pll)
logq <- -0.5 * log(gpdN.infomat(par = para, dat = dat, method = "obs", N = N)[-1,
-1]) + qmlecontrib + log(abs(det(rbind(c(c(phi.mle) - gpdN.phi(par = para,
dat = dat, V = V, N = N)), gpdN.dphi(par = para, dat = dat, V = V, N = N)[-1,
]))))
rs + logq - log(sqrt(abs(rs)))
}
tem.max <- optim(par = mle[1], fn = tem.max.opt, method = "Brent", dat = dat, lower = max(1e-05,
mle[1] - std.error), upper = mle[1] + std.error, control = list(abstol = 1e-10))$par
}
if ("modif" %in% mod) {
# Tangent exponential model approximation of Fraser and Reid to the profile likelihood
tem.objfunc.Nmean <- function(par) {
0.5 * log(gpdN.infomat(par = par, dat = dat, method = "obs", N = N)[2, 2]) -
log(abs(gpdN.dphi(par = par, dat = dat, N = N, V = V[, 2, drop = FALSE])[2, 1]))
}
optim.tem.fn.Nmean <- function(psi) {
theta.psi.opt <- constr.mle.Nmean(psi)
param <- c(psi, theta.psi.opt)
ll <- gpdN.ll(param, dat = dat, N = N)
ll + tem.objfunc.Nmean(param)
}
# TEM profile log likelihood values for psi
proflltem <- profll + suppressWarnings(apply(pars, 1, tem.objfunc.Nmean))
# Maximum objective function for TEM
tem.mle.opt <- optim(par = mle[1], fn = optim.tem.fn.Nmean, method = "Brent", lower = max(1e-05,
mle[1] - std.error), upper = mle[1] + std.error, control = list(fnscale = -1))
tem.mle <- c(tem.mle.opt$par, constr.mle.Nmean(tem.mle.opt$par))
# Severini empirical covariance function adjustment to profile likelihood Score function -
# observation-wise for xi
gpdN.score.f <- function(par, dat, N) {
z = par[1]
xi = par[2]
cst <- exp(lgamma(N + 1) + lgamma(1 - xi) - lgamma(N + 1 - xi))
-(psigamma(N - xi + 1) * cst - psigamma(-xi + 1) * cst) * dat * (1/xi + 1)/(z *
(dat * (cst - 1)/z + 1)) + ((psigamma(N - xi + 1) * cst - psigamma(-xi +
1) * cst) * xi * z/(cst - 1)^2 - z/(cst - 1)) * (cst - 1)/(xi * z) + log(dat *
(cst - 1)/z + 1)/xi^2
}
# Score at MLE (sums to zero)
score.Nmean.mle <- gpdN.score.f(mle, dat, N)
empcov.objfunc.Nmean <- function(par) {
0.5 * log(gpdN.infomat(par = par, dat = dat, method = "obs", N = N)[2, 2]) -
log(abs(sum(score.Nmean.mle * gpdN.score.f(par, dat, N = N))))
}
profllempcov <- profll + suppressWarnings(apply(pars, 1, empcov.objfunc.Nmean))
optim.empcov.fn.Nmean <- function(psi) {
theta.psi.opt <- constr.mle.Nmean(psi)
param <- c(psi, theta.psi.opt)
ll <- gpdN.ll(param, dat = dat, N = N)
ll + empcov.objfunc.Nmean(param)
}
empcov.mle.opt <- optim(par = mle[1], fn = optim.empcov.fn.Nmean, method = "Brent",
lower = max(1e-05, mle[1] - std.error), upper = mle[1] + std.error, control = list(fnscale = -1))
empcov.mle <- c(empcov.mle.opt$par, constr.mle.Nmean(empcov.mle.opt$par))
}
}
# Return profile likelihood and quantities of interest (modified likelihoods)
colnames(pars) <- names(mle)
ans <- list(mle = mle, pars = pars, psi.max = as.vector(mle[param]), param = param, std.error = std.error,
psi = psi, pll = profll, maxpll = maxll, r = r)
# Shift by threshold if non-null
if(shiftres){
ans$psi <- ans$psi + threshold
ans$mle[1] <- ans$psi.max <- ans$mle[1] + threshold
ans$pars[,1] <- ans$pars[,1] + threshold
}
if ("tem" %in% mod) {
ans$q <- qcor
ans$rstar <- rstar
ans$tem.psimax <- as.vector(tem.max) + ifelse(shiftres, threshold, 0)
ans$normal <- c(ans$psi.max, ans$std.error)
if (correction && length(psi) > 10) {
ans <- spline.corr(ans)
}
}
if ("modif" %in% mod) {
ans$tem.mle <- ifelse(param == "shape", tem.mle[2], tem.mle[1])
if(shiftres){ ans$tem.mle[1] <- ans$tem.mle[1] + threshold }
ans$tem.pll <- proflltem
ans$tem.maxpll <- as.vector(tem.mle.opt$value)
ans$empcov.mle <- ifelse(param == "shape", empcov.mle[2], empcov.mle[1])
if(shiftres){ ans$empcov.mle[1] <- ans$empcov.mle[1] + threshold }
ans$empcov.pll <- as.vector(profllempcov)
ans$empcov.maxpll <- as.vector(empcov.mle.opt$value)
}
if ("tem" %in% mod) {
class(ans) <- c("eprof", "fr")
} else {
class(ans) <- "eprof"
}
ans$family <- "gpd"
ans$threshold <- threshold
if(plot){
plot(ans)
}
return(invisible(ans))
}
#' Plot of tangent exponential model profile likelihood
#'
#' This function is adapted from the \code{plot.fr} function from the \code{hoa} package bundle.
#' It differs from the latter mostly in the placement of legends.
#'
#' @param x an object of class \code{fr} returned by \code{\link{gpd.tem}} or \code{\link{gev.tem}}.
#' @param ... further arguments to \code{plot} currently ignored. Providing a numeric vector \code{which} allows for custom selection of the plots. A logical \code{all}. See \strong{Details}.
#' @return graphs depending on argument \code{which}
#' @details Plots produced depend on the integers provided in \code{which}. \code{1} displays the Wald pivot, the likelihood root \code{r}, the modified likelihood root \code{rstar} and the likelihood modification \code{q} as functions of the parameter \code{psi}. \code{2} gives the renormalized profile log likelihood and adjusted form, with the maximum likelihood having ordinate value of zero. \code{3} provides the significance function, a transformation of \code{1}. Lastly, \code{4} plots the correction factor as a function of the likelihood root; it is a diagnostic plot aimed for detecting failure of
#' the asymptotic approximation, often due to poor numerics in a neighborhood of \code{r=0}; the function should be smooth. The function \code{\link{spline.corr}} is designed to handle this by correcting numerically unstable estimates, replacing outliers and missing values with the fitted values from the fit.
#'
#'
#' @references Brazzale, A. R., Davison, A. C. and Reid, N. (2007). \emph{Applied Asymptotics: Case Studies in Small-Sample Statistics}. Cambridge University Press, Cambridge.
#' @export
plot.fr <- function(x, ...) {
# plot a fraser-reid object
whichPlot <- c(1:4) #default
if (length(list(...)) > 0) {
if ("which" %in% names(list(...))) {
whichPlot <- list(...)$which
whichPlot <- (1:4)[c(1:4 %in% whichPlot)]
} else if ("all" %in% names(list(...))) {
if (!is.logical(all)) {
stop("Invalid \"all\" parameter")
}
if (list(...)$all) {
whichPlot <- 1:4
} else {
whichPlot <- 1:2
}
}
}
old.pars <- par(no.readonly = TRUE)
if (sum(c(1, 2, 3, 4) %in% whichPlot) > 2) {
par(mfrow = c(2, 2), mar = c(4.8, 4.8, 1, 0.1))
} else if (sum(c(1, 2, 3, 4) %in% whichPlot) == 2) {
par(mfrow = c(1, 2))
}
fr <- x
xl <- ifelse(is.null(fr$param), expression(psi), fr$param)
if (1 %in% whichPlot) {
plot(fr$psi, fr$r, type = "l", xlab = xl, ylab = "Value of pivot", ylim = c(-4, 4),
panel.first = abline(h = qnorm(c(0.005, 0.025, 0.05, 0.5, 0.95, 0.975, 0.995)),
col = "grey", lwd = 0.7), bty = "l")
lines(fr$psi, (fr$normal[1] - fr$psi)/fr$normal[2], col = "green", lwd = 1.5)
lines(fr$psi, fr$q, col = "red", lwd = 1.5)
lines(fr$psi, fr$r, lwd = 1.5)
lines(fr$psi, fr$rstar, col = "blue")
legend(x = "topright", c("Wald pivot", "lik. root", "modif. root", expression(q(psi))),
lty = c(1, 1, 1, 1), x.intersp = 0.2, lwd = 1.5, seg.len = 0.5, col = c("green",
"black", "blue", "red"), bty = "n", cex = 0.9, xjust = 1)
}
# top right: log likelihood (and adjusted version, I think?) as a function of psi
if (2 %in% whichPlot) {
plot(fr$psi, -fr$r^2/2, type = "l", xlab = xl, ylab = "Profile log likelihood", ylim = c(-8,
0), panel.first = abline(h = -qchisq(c(0.95, 0.99), df = 1)/2, col = "grey", lwd = 0.7),
lwd = 1.5, bty = "l")
lines(fr$psi, -fr$rstar^2/2, col = "blue", lwd = 1.5)
legend(x = "bottomright", c("profile","tem"),
lty = c(1, 1), x.intersp = 0.2, lwd = 1.5, seg.len = 0.5, col = c("black", "blue"),
bty = "n")
# optional: add diagnostic panels
}
if (3 %in% whichPlot) {
# lower left: plot of Phi(pivot) as a function of psi
plot(fr$psi, pnorm(fr$r), type = "l", xlab = xl, ylab = "Significance function", ylim = c(0,
1), panel.first = abline(h = c(0.025, 0.05, 0.5, 0.95, 0.975), col = "grey", lwd = 0.7),
lwd = 1.5, bty = "l")
lines(fr$psi, pnorm(fr$q), col = "red", lwd = 1.5)
lines(fr$psi, pnorm(fr$rstar), col = "blue", lwd = 1.5)
legend(x = "topright", c("lik. root", "modif. root", expression(q(psi))), lty = c(1,
1, 1), col = c("black", "blue", "red"), bty = "n", x.intersp = 0.2, lwd = 1.5,
seg.len = 0.5, cex = 0.9)
}
# lower right: log(q/r)/r as a function of r (should be smooth)
if (4 %in% whichPlot) {
fit.r <- stats::smooth.spline(x = na.omit(cbind(fr$r, fr$rstar)), cv = FALSE)
pr <- predict(fit.r, 0)$y
plot(fr$r, fr$rstar-fr$r, type = "l", xlab = "Likelihood root r", ylab = expression(paste("Correction factor log(q/r)/r")),
panel.first = {
abline(h = 0, col = "grey", lwd = 0.7)
abline(v = 0, col = "grey", lwd = 0.7)
abline(v = pr, col = "grey", lwd = 0.7, lty = 2)
}, lwd = 1.5, bty = "l")
}
par(old.pars)
}
#' Spline correction for Fraser-Reid approximations
#'
#' The tangent exponential model can be numerically unstable for values close to \eqn{r = 0}.
#' This function corrects these incorrect values, which are interpolated using splines.
#' The function takes as input an object of class \code{fr} and returns the same object with
#' different \code{rstar} values.
#' @section Warning:
#'
#' While penalized (robust) splines often do a good job at capturing and correcting for numerical outliers and \code{NA}, it
#' may also be driven by unusual values lying on the profile log-likelihood the curve or fail to detect outliers (or falsely identifying `correct' values as outliers). The user should always validate by comparing the plots of both the uncorrected (raw) output of the object with that of \code{spline.corr}.
#' @details If available, the function uses \code{cobs} from the eponym package. The latter handles constraints and smoothness penalties, and is more robust than the equivalent \code{\link[stats]{smooth.spline}}.
#'
#' @param fr an object of class \code{fr}, normally the output of \link{gpd.tem} or \link{gev.tem}.
#' @param method string for the method, either \code{cobs} (constrained robust B-spline from eponym package) or \code{smooth.spline}
#' @return an object of class \code{fr}, containing as additional arguments \code{spline} and a modified \code{rstar} argument.
#' @keywords internal
#' @export
spline.corr <- function(fr, method = c("cobs","smooth.spline")) {
# Step 1: fit a smoothing spline to rstar If fit failed for some values (for example when
# shape forced to be < 1) Remove those values
method <- match.arg(method[1], choices = c("cobs","smooth.spline"))
if (all(is.nan(fr$q)) || all(is.nan(fr$rstar))) {
# If could not compute Fraser-Reid correction, abort
return(fr)
}
fitfailed <- which(!is.finite(fr$r))
if (length(fitfailed) > 0) {
fr$r <- fr$r[-fitfailed]
fr$rstar <- fr$rstar[-fitfailed]
fr$q <- fr$q[-fitfailed]
fr$psi <- fr$psi[-fitfailed]
}
w <- pchisq(fr$r^2, 0.5)
# If any correction for q failed and returned NA
corfailed <- which(!is.finite(fr$rstar))
# If equispaced values for psi between MLE and other, then we have r = 0
corfailed <- c(corfailed, which(fr$r == 0))
if (length(corfailed) > 0) {
resp <- (fr$rstar - fr$r)[-corfailed]
regr <- fr$r[-corfailed]
w <- w[-corfailed]
} else {
resp <- (fr$rstar - fr$r)
regr <- fr$r
}
if (method == "cobs" && requireNamespace("cobs", quietly = TRUE)) {
spline <- cobs::cobs(y = resp, x = regr, w = w, constraint = "none", lambda = 1, nknots = 20,
print.mesg = FALSE, print.warn = FALSE)$fitted
} else {
spline <- rev(stats::smooth.spline(y = resp, x = regr, w = w, spar = 0.9, cv = TRUE)$y)
}
# Compute difference between fitted values and rstar
departure <- spline - resp
# Ad-hoc fix of the values close to MLE where the numerical precision causes difficulty
# Outlier detection via chi-square test From package outliers, (c)Lukasz Komsta
scores <- function(x, prob = NA) {
(x - mean(x))^2/var(x) > qchisq(prob, 1)
}
bad <- which(scores(departure, prob = 0.95))
if (length(bad) > 0) {
# Exclude those values if they are in the end of the distribution
bad <- bad[intersect(which(bad < 0.85 * length(departure)), which(bad > 0.15 * length(departure)))]
# Remove outliers and fit again (with less smoothness)
}
if (length(bad) > 0) {
resp[bad] <- NA
w <- w[-bad]
}
if (requireNamespace("cobs", quietly = TRUE)) {
spline <- cobs::cobs(y = resp, x = regr, constraint = "none", w = w, lambda = -1, ic = "SIC",
knots.add = TRUE, repeat.delete.add = TRUE, print.mesg = FALSE, print.warn = FALSE)
fr$spline <- spline
fr$rstar <- predict(spline, fr$r, interval = "none")[, 2] + fr$r
} else {
spline <- stats::smooth.spline(x = na.omit(cbind(regr, resp)), cv = FALSE, all.knots = TRUE)
fr$spline <- spline
fr$rstar <- predict(spline, fr$r)$y + fr$r
}
return(fr)
}
#' Bridging the singularity for higher order asymptotics
#'
#' The correction factor \eqn{\log(q/r)/r} for the
#' likelihood root is unbounded in the vincinity of
#' the maximum likelihood estimator. The thesis of
#' Rongcai Li (University of Toronto, 2001)
#' explores different ways of bridging this
#' singularity, notably using asymptotic expansions.
#'
#' The poor man's method used here consists in
#' fitting a robust regression to \eqn{1/q-1/r}
#' as a function of \eqn{r} and using predictions
#' from the model to solve for \eqn{q}. This
#' approach is seemingly superior to that
#' previously used in \link{spline.corr}.
#'
#' @param fran object of class \code{fr}
#' @param print.warning logical; should warning message be printed? Default to \code{FALSE}
##' @return an object of class \code{fr}, containing as additional arguments \code{spline} and a modified \code{rstar} argument.
#' @keywords internal
#' @export
tem.corr <- function(fr, print.warning = FALSE) {
if(!requireNamespace("MASS", quietly = TRUE)){
stop("The \"MASS\" package is required for this function to work")
}
if (all(is.nan(fr$q)) || all(is.nan(fr$rstar))) {
# If could not compute Fraser-Reid correction, abort
return(fr)
}
fitfailed <- which(!is.finite(fr$r))
if (length(fitfailed) > 0) {
fr$r <- fr$r[-fitfailed]
fr$rstar <- fr$rstar[-fitfailed]
fr$q <- fr$q[-fitfailed]
fr$psi <- fr$psi[-fitfailed]
}
if(length(fr$psi) < 25L){
if(print.warning){
warning("The correction for the tangent exponential model\n is based on less than 25 observations.")
}
}
# this is approximately linear for fixed data
resp <- 1/fr$q - 1/fr$r
# fit a robust regression to discount observations that are outlying
robust_reg <- MASS::rlm(resp ~ fr$r)
# Replace q values by predictions
pred <- predict(robust_reg)
qhat <- 1/(pred + 1/fr$r)
fr$q_old <- fr$q
fr$rstar_old <- fr$rstar
fr$q <- qhat
fr$rstar <- fr$r + log(fr$q/fr$r)/fr$r
fr$tem.psimax
return(fr)
}
#' Tangent exponential model approximation for the GEV distribution
#'
#' The function \code{gev.tem} provides a tangent exponential model (TEM) approximation
#' for higher order likelihood inference for a scalar parameter for the generalized extreme value distribution.
#' Options include location scale and shape parameters as well as value-at-risk (or return levels).
#' The function attempts to find good values for \code{psi} that will
#' cover the range of options, but the fail may fit and return an error.
#'
#'
#' @param param parameter over which to profile
#' @param psi scalar or ordered vector of values for the interest parameter. If \code{NULL} (default), a grid of values centered at the MLE is selected
#' @param N size of block over which to take maxima. Required only for \code{param} \code{Nmean} and \code{Nquant}.
#' @param p tail probability for the (1-p)th quantile (return levels). Required only if \code{param = 'retlev'}
#' @param q probability level of quantile. Required only for \code{param} \code{Nquant}.
#' @param dat sample vector for the GEV distribution
#' @param n.psi number of values of \code{psi} at which the likelihood is computed, if \code{psi} is not supplied (\code{NULL}). Odd values are more prone to give rise to numerical instabilities near the MLE. If \code{psi} is a vector of length 2 and \code{n.psi} is greater than 2, these are taken to be endpoints of the sequence.
#' @param plot logical indicating whether \code{plot.fr} should be called upon exit
#' @param correction logical indicating whether \link{spline.corr} should be called.
#' @author Leo Belzile
#' @return an invisible object of class \code{fr} (see \code{tem} in package \code{hoa}) with elements
#' \itemize{
#' \item{\code{normal}: }{maximum likelihood estimate and standard error of the interest parameter \eqn{psi}}
#' \item{\code{par.hat}: }{maximum likelihood estimates}
#' \item{\code{par.hat.se}: }{standard errors of maximum likelihood estimates}
#' \item{\code{th.rest}: }{estimated maximum profile likelihood at (\eqn{psi}, \eqn{\hat{\lambda}})}
#' \item{\code{r}: }{values of likelihood root corresponding to \eqn{\psi}}
#' \item{\code{psi}: }{vector of interest parameter}
#' \item{\code{q}: }{vector of likelihood modifications}
#' \item{\code{rstar}: }{modified likelihood root vector}
#' \item{\code{rstar.old}: }{uncorrected modified likelihood root vector}
#' \item{\code{param}: }{parameter}
#' }
#' @export
#' @examples
#' \dontrun{
#' set.seed(1234)
#' dat <- rgev(n = 40, loc = 0, scale = 2, shape = -0.1)
#' gev.tem('shape', dat = dat, plot = TRUE)
#' gev.tem('quant', dat = dat, p = 0.01, plot = TRUE)
#' gev.tem('scale', psi = seq(1, 4, by = 0.1), dat = dat, plot = TRUE)
#' dat <- rgev(n = 40, loc = 0, scale = 2, shape = 0.2)
#' gev.tem('loc', dat = dat, plot = TRUE)
#' gev.tem('Nmean', dat = dat, p = 0.01, N=100, plot = TRUE)
#' gev.tem('Nquant', dat = dat, q = 0.5, N=100, plot = TRUE)
#' }
gev.tem <- function(param = c("loc", "scale", "shape", "quant", "Nmean", "Nquant"), dat, psi = NULL,
p = NULL, q = 0.5, N = NULL, n.psi = 50, plot = TRUE, correction = TRUE) {
if (param %in% c("VaR", "retlev"))
{
param <- "quant"
} #Compatibility following change of notation 13/07/2017
if (param == "scale" && !is.null(psi)) {
if (isTRUE(any(psi < 0))) {
stop("Invalid argument: scale parameter must be positive")
}
}
tem_out <- gev.pll(psi = psi, param = param, mod = "tem", dat = dat, N = N, p = p, q = q,
correction = correction)
if (plot) {
plot.fr(tem_out)
}
return(invisible(tem_out))
}
#' Tangent exponential model approximation for the GP distribution
#'
#' The function \code{gpd.tem} provides a tangent exponential model (TEM) approximation
#' for higher order likelihood inference for a scalar parameter for the generalized Pareto distribution. Options include
#' scale and shape parameters as well as value-at-risk (also referred to as quantiles, or return levels)
#' and expected shortfall. The function attempts to find good values for \code{psi} that will
#' cover the range of options, but the fit may fail and return an error. In such cases, the user can try to find good
#' grid of starting values and provide them to the routine.
#'
#' As of version 1.11, this function is a wrapper around \code{gpd.pll}.
#'
#' @details The interpretation for \code{m} is as follows: if there are on average \eqn{m_y} observations per year above the threshold, then \eqn{m = Tm_y} corresponds to \eqn{T}-year return level.
#'
#' @param param parameter over which to profile
#' @param threshold threshold value corresponding to the lower bound of the support or the location parameter of the generalized Pareto distribution.
#' @param psi scalar or ordered vector of values for the interest parameter. If \code{NULL} (default), a grid of values centered at the MLE is selected. If \code{psi} is of length 2 and \code{n.psi}>2, it is assumed to be the minimal and maximal values at which to evaluate the profile log likelihood.
#' @param m number of observations of interest for return levels. See \strong{Details}. Required only for \code{param = 'VaR'} or \code{param = 'ES'}.
#' @param N size of block over which to take maxima. Required only for \code{args} \code{Nmean} and \code{Nquant}.
#' @param p tail probability, equivalent to \eqn{1/m}. Required only for \code{args} \code{quant}.
#' @param q level of quantile for N-block maxima. Required only for \code{args} \code{Nquant}.
#' @param dat sample vector for the GP distribution
#' @param n.psi number of values of \code{psi} at which the likelihood is computed, if \code{psi} is not supplied (\code{NULL}). Odd values are more prone to give rise to numerical instabilities near the MLE
#' @param plot logical indicating whether \code{plot.fr} should be called upon exit
#' @param correction logical indicating whether \link{spline.corr} should be called.
#' @author Leo Belzile
#' @return an invisible object of class \code{fr} (see \code{tem} in package \code{hoa}) with elements
#' \itemize{
#' \item{\code{normal}: }{maximum likelihood estimate and standard error of the interest parameter \eqn{psi}}
#' \item{\code{par.hat}: }{maximum likelihood estimates}
#' \item{\code{par.hat.se}: }{standard errors of maximum likelihood estimates}
#' \item{\code{th.rest}: }{estimated maximum profile likelihood at (\eqn{psi}, \eqn{\hat{\lambda}})}
#' \item{\code{r}: }{values of likelihood root corresponding to \eqn{\psi}}
#' \item{\code{psi}: }{vector of interest parameter}
#' \item{\code{q}: }{vector of likelihood modifications}
#' \item{\code{rstar}: }{modified likelihood root vector}
#' \item{\code{rstar.old}: }{uncorrected modified likelihood root vector}
#' \item{\code{param}: }{parameter}
#' }
#' @export
#' @examples
#' set.seed(123)
#' dat <- rgp(n = 40, scale = 1, shape = -0.1)
#' #with plots
#' m1 <- gpd.tem(param = 'shape', n.psi = 50, dat = dat, plot = TRUE)
#' \dontrun{
#' m2 <- gpd.tem(param = 'scale', n.psi = 50, dat = dat)
#' m3 <- gpd.tem(param = 'VaR', n.psi = 50, dat = dat, m = 100)
#' #Providing psi
#' psi <- c(seq(2, 5, length = 15), seq(5, 35, length = 45))
#' m4 <- gpd.tem(param = 'ES', dat = dat, m = 100, psi = psi, correction = FALSE)
#' mev:::plot.fr(m4, which = c(2, 4))
#' plot(fr4 <- spline.corr(m4))
#' confint(m1)
#' confint(m4, parm = 2, warn = FALSE)
#' m5 <- gpd.tem(param = 'Nmean', dat = dat, N = 100, psi = psi, correction = FALSE)
#' m6 <- gpd.tem(param = 'Nquant', dat = dat, N = 100, q = 0.7, correction = FALSE)
#' }
gpd.tem <- function(dat, param = c("scale", "shape", "quant", "VaR", "ES", "Nmean", "Nquant"),
psi = NULL, m = NULL, threshold = 0, n.psi = 50, N = NULL, p = NULL, q = NULL, plot = FALSE,
correction = TRUE) {
if (param %in% c("VaR", "ES") && is.null(m)) {
stop("Parameter \"m\" missing")
}
dat <- dat - threshold
tem <- gpd.pll(psi = psi, param = param, mod = "tem", dat = dat, N = N, m = m, mle = NULL,
q = q, p = p, correction = correction)
if (plot) {
plot.fr(tem)
}
return(invisible(tem))
}
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Defines functions gpd.tem gev.tem tem.corr spline.corr plot.fr gpd.pll gev.pll plot.eprof confint.eprof
Documented in confint.eprof gev.pll gev.tem gpd.pll gpd.tem plot.eprof plot.fr spline.corr tem.corr
# This function is added to ensure that the local solver can be fixed to something different
# than COBYLA - see https://github.com/jyypma/nloptr/pull/38 and
# https://github.com/stevengj/nlopt/issues/118
# .auglag <- function(x0, fn, gr = NULL, lower = NULL, upper = NULL, hin = NULL, hinjac = NULL,
# heq = NULL, heqjac = NULL, localsolver = c("COBYLA"), localtol = 1e-06, ineq2local = FALSE,
# nl.info = FALSE, control = list(), ...) {
# if (ineq2local) {
# stop("Inequalities to local solver: feature not yet implemented.")
# }
# localsolver <- toupper(localsolver)
# if (localsolver %in% c("COBYLA", "BOBYQA")) {
# # changed this line to add BOBYQA local solver
# dfree <- TRUE
# gsolver <- "NLOPT_LN_AUGLAG"
# lsolver <- paste("NLOPT_LN_", localsolver, sep = "")
# } else if (localsolver %in% c("LBFGS", "MMA", "SLSQP")) {
# dfree <- FALSE
# gsolver <- "NLOPT_LD_AUGLAG"
# lsolver <- paste("NLOPT_LD_", localsolver, sep = "")
# } else {
# stop("Only local solvers allowed: BOBYQA, COBYLA, LBFGS, MMA, SLSQP.")
# }
# .fn <- match.fun(fn)
# fn <- function(x) .fn(x, ...)
# if (!dfree && is.null(gr)) {
# gr <- function(x) nloptr::nl.grad(x, fn)
# }
# opts <- nloptr::nl.opts(control)
# opts$algorithm <- gsolver
# local_opts <- list(algorithm = lsolver, xtol_rel = localtol, eval_grad_f = if (!dfree) gr else NULL)
# opts$local_opts <- local_opts
# if (!is.null(hin)) {
# .hin <- match.fun(hin)
# hin <- function(x) (-1) * .hin(x)
# }
# if (!dfree) {
# if (is.null(hinjac)) {
# hinjac <- function(x) nloptr::nl.jacobian(x, hin)
# } else {
# .hinjac <- match.fun(hinjac)
# hinjac <- function(x) (-1) * .hinjac(x)
# }
# }
# if (!is.null(heq)) {
# .heq <- match.fun(heq)
# heq <- function(x) .heq(x)
# }
# if (!dfree) {
# if (is.null(heqjac)) {
# heqjac <- function(x) nloptr::nl.jacobian(x, heq)
# } else {
# .heqjac <- match.fun(heqjac)
# heqjac <- function(x) .heqjac(x)
# }
# }
# S0 <- nloptr::nloptr(x0, eval_f = fn, eval_grad_f = gr, lb = lower, ub = upper, eval_g_ineq = hin,
# eval_jac_g_ineq = hinjac, eval_g_eq = heq, eval_jac_g_eq = heqjac, opts = opts)
# if (nl.info)
# print(S0)
# S1 <- list(par = S0$solution, value = S0$objective, iter = S0$iterations, global_solver = gsolver,
# local_solver = lsolver, convergence = S0$status, message = S0$message)
# return(S1)
# }
#' Confidence intervals for profile likelihood objects
#'
#' Computes confidence intervals for the parameter psi for profile likelihood objects.
#' This function uses spline interpolation to derive \code{level} confidence intervals
#'
#' @param object an object of class \code{eprof}, normally the output of \link{gpd.pll} or \link{gev.pll}.
#' @param parm a specification of which parameters are to be given confidence intervals,
#' either a vector of numbers or a vector of names. If missing, all parameters are considered.
#' @param level confidence level, with default value of 0.95
#' @param prob percentiles, with default giving symmetric 95\% confidence intervals
#' @param method string for the method, either \code{cobs} (constrained robust B-spline from eponym package) or \code{smooth.spline}
#' @param ... additional arguments passed to functions. Providing a logical \code{warn=FALSE} turns off warning messages when the lower or upper confidence interval for \code{psi} are extrapolated beyond the provided calculations.
#' @param print should a summary be printed. Default to \code{FALSE}.
#' @return returns a 2 by 3 matrix containing point estimates, lower and upper confidence intervals based on the likelihood root and modified version thereof
#' @export
#' @importFrom Rsolnp solnp
#' @importFrom alabama auglag
#' @importFrom utils tail
confint.eprof <- function(object, parm, level = 0.95, prob = c((1-level)/2, 1-(1-level)/2), print = FALSE, method = c("cobs","smooth.spline"), ...) {
if(!isTRUE(all.equal(diff(prob),level, check.attributes = FALSE))){
warning("Incompatible arguments: \"level\" does not match \"prob\".")
}
method <- match.arg(method[1], c("cobs","smooth.spline"))
args <- list(...)
if ("warn" %in% names(args) && is.logical(args$warn)) {
warn <- args$warn
} else {
warn <- TRUE
}
if (length(prob) != 2) {
stop("\"prob\" must be a vector of size 2")
prob <- sort(prob)
}
if (missing(parm)) {
parm <- NULL
ind <- args$ind
if (!is.null(object$pll) || !is.null(object$r)) {
parm <- c(parm, "profile")
ind <- c(ind, 1)
}
if (!is.null(object$rstar)) {
parm <- c(parm, "tem")
ind <- c(ind, 2)
}
if (!is.null(object$tem.pll)) {
parm <- c(parm, "modif.tem")
ind <- c(ind, 3)
}
if (!is.null(object$empcov.pll)) {
parm <- c(parm, "modif.empcov")
ind <- c(ind, 4)
}
} else {
if (is.numeric(parm)) {
ind <- parm
parm <- c("profile", "tem", "modif.tem", "modif.empcov")[ind]
} else {
parm <- match.arg(arg = parm, choices = c("profile", "tem", "modif.tem", "modif.empcov",
"r", "rstar"), several.ok = TRUE)
parm[parm %in% "r"] <- "profile"
parm[parm %in% "rstar"] <- "tem"
ind <- which(c("profile", "tem", "modif.tem", "modif.empcov") %in% parm)
}
parm <- unique(parm)
ind <- unique(ind[ind %in% 1:4])
}
if (length(ind) == 0) {
stop("Invalid \"parm\" argument.")
}
qulev <- qnorm(1-prob)
conf <- matrix(ncol = 4, nrow = 3)
for (i in ind) {
if (i == 1) {
if (is.null(object$pll) && is.null(object$r)) {
break
}
if (is.null(object$r)) {
# no r object, but must have pll + maxpll
object$r <- sign(object$psi.max - object$psi) * sqrt(2 * (object$maxpll - object$pll))
} else{
object$r[is.infinite(object$r)] <- NA
}
if (is.null(object$normal)) {
object$normal <- c(object$psi.max, object$std.error)
}
if (method == "cobs" && requireNamespace("cobs", quietly = TRUE)) {
fit.r <- cobs::cobs(x = object$r, y = object$psi, constraint = "decrease",
lambda = 0, ic = "SIC", pointwise = cbind(0, 0, object$normal[1]), knots.add = TRUE,
repeat.delete.add = TRUE, print.mesg = FALSE, print.warn = FALSE)
pr <- predict(fit.r, c(0, qulev))[,2]
} else {
fit.r <- stats::smooth.spline(x = na.omit(cbind(object$r, object$psi)), cv = FALSE)
pr <- predict(fit.r, c(0, qulev))$y
pr[1] <- object$normal[1]
}
conf[,i] <- pr
if (warn) {
if (!any(object$r > qnorm(prob[1]))) {
warning("Extrapolating the lower confidence interval for the profile likelihood ratio test")
}
if (!any(object$r < qnorm(prob[2]))) {
warning("Extrapolating the upper confidence interval for the profile likelihood ratio test")
}
}
} else if (i == 2) {
if (is.null(object$rstar)) {
break
}
if (method == "cobs" && requireNamespace("cobs", quietly = TRUE)) {
fit.rst <- cobs::cobs(x = object$rstar, y = object$psi, constraint = "decrease",
lambda = 0, ic = "SIC", knots.add = TRUE, repeat.delete.add = TRUE, print.mesg = FALSE,
print.warn = FALSE)
prst <- predict(fit.rst, c(0, qulev))[,2]
} else {
fit.rst <- stats::smooth.spline(x = na.omit(cbind(object$rstar, object$psi)),
cv = FALSE)
prst <- predict(fit.rst, c(0, qulev))$y
}
if (!is.null(object$tem.psimax)) {
prst[1] <- object$tem.psimax
} else{
object$tem.psimax <- prst[1]
}
conf[,i] <- prst
# lines(x=object$rstar,fit.rst$fitted,col=2,pch=19)
if (warn) {
if (!any(object$rstar > qulev[2])) {
warning("Extrapolating the adjusted lower confidence interval for rstar.")
}
if (!any(object$rstar < qulev[1])) {
warning("Extrapolating the adjusted upper confidence interval for rstar")
}
}
} else if (i == 3) {
if (is.null(object$tem.pll)) {
break
}
if (method == "cobs" && requireNamespace("cobs", quietly = TRUE)) {
fit.mtem <- cobs::cobs(x = sign(object$tem.mle - object$psi) * sqrt(-2 * (object$tem.pll -
object$tem.maxpll)), y = object$psi, constraint = "decrease", lambda = 0,
ic = "SIC", knots.add = TRUE, repeat.delete.add = TRUE, print.mesg = FALSE,
print.warn = FALSE)
ptem <- predict(fit.mtem, c(0, qulev))[, 2]
} else {
fit.mtem <- stats::smooth.spline(x = na.omit(cbind(sign(object$tem.mle - object$psi) *
sqrt(-2 * (object$tem.pll - object$tem.maxpll)), object$psi)), cv = FALSE)
ptem <- predict(fit.mtem, c(0, qulev))$y
}
ptem[1] <- object$tem.mle
conf[,i] <- ptem
#TODO add warnings about extrapolation
} else if (i == 4) {
if (is.null(object$empcov.pll)) {
break
}
if (method == "cobs" && requireNamespace("cobs", quietly = TRUE)) {
fit.mempcov <- cobs::cobs(x = sign(object$empcov.mle - object$psi) * sqrt(-2 *
(object$empcov.pll - object$empcov.maxpll)), y = object$psi, constraint = "decrease",
lambda = 0, ic = "SIC", knots.add = TRUE, repeat.delete.add = TRUE, print.mesg = FALSE,
print.warn = FALSE)
pempcov <- predict(fit.mempcov, c(0.5, qulev))[, 2]
} else {
fit.mempcov <- stats::smooth.spline(x = na.omit(cbind(sign(object$empcov.mle -
object$psi) * sqrt(-2 * (object$empcov.pll - object$empcov.maxpll)), object$psi)),
cv = FALSE)
pempcov <- predict(fit.mempcov, c(0.5, qulev))$y
}
pempcov[1] <- object$empcov.mle
conf[,i] <- pempcov
}
}
if (!is.null(conf)) {
colnames(conf) <- c("Profile", "TEM", "Severini (TEM)", "Severini (emp. cov.)")
rownames(conf) <- c("Estimate", "Lower CI", "Upper CI")
#Check output makes sense - lower CI is lower than estimate
#and similarly upper CI is above estimate
wrong_below <- which(conf[2,] > conf[1,])
if(length(wrong_below)>0){
conf[2,][wrong_below] <- NA
}
wrong_above <- which(conf[3,] < conf[1,])
if(length(wrong_above)>0){
conf[3,][wrong_above] <- NA
}
if(print){
cat("Point estimate for the parameter of interest psi:\n")
cat("Maximum likelihood :", round(object$psi.max, 3), "\n")
if(2 %in% ind){
cat("Tangent exponential model :", round(object$tem.psimax, 3), "\n")
}
if(3 %in% ind){
cat("Severini's profile (TEM) :", round(object$tem.mle, 3), "\n")
}
if(4 %in% ind){
cat("Severini's profile (empcov) :", round(object$empcov.mle, 3), "\n")
}
cat("\n")
cat("Confidence intervals, levels :", prob, "\n")
cat("Wald intervals :", round(object$psi.max+sort(qulev)*object$std.error, 3), "\n")
cat("Profile likelihood :", round(conf[2:3,1], 3), "\n")
if(2 %in% ind){
cat("Tangent exponential model :", round(conf[2:3,2], 3), "\n")
}
if(3 %in% ind){
cat("Severini's profile (TEM) :", round(conf[2:3,3], 3), "\n")
}
if(4 %in% ind){
cat("Severini's profile (empcov) :", round(conf[2:3,4], 3), "\n")
}
}
return(invisible(conf[,ind]))
}
}
#' Plot of (modified) profile likelihood
#'
#' The function plots the (modified) profile likelihood and the tangent exponential profile likelihood
#'
#' @param x an object of class \code{eprof} returned by \code{\link{gpd.pll}} or \code{\link{gev.pll}}.
#' @param ... further arguments to \code{plot}.
#' @return a graph of the (modified) profile likelihoods
#' @references Brazzale, A. R., Davison, A. C. and Reid, N. (2007). \emph{Applied Asymptotics: Case Studies in Small-Sample Statistics}. Cambridge University Press, Cambridge.
#' @references Severini, T. A. (2000). \emph{Likelihood Methods in Statistics}. Oxford University Press, Oxford.
#' @export
plot.eprof <- function(x, ...) {
# plot the profile log-likelihoods
#old.pars <- par(no.readonly = TRUE)
args <- list(...)
lik <- list()
if (is.null(x$pll) && !is.null(x$r)) {
lik$npll <- -x$r^2/2
} else if (!is.null(x$pll)) {
lik$npll <- x$pll - x$maxpll
} else {
stop("Invalid object provided")
}
if (!is.null(x$tem.pll)) {
lik$tem.npll <- x$tem.pll - x$tem.maxpll
}
if (!is.null(x$empcov.pll)) {
lik$empcov.npll <- x$empcov.pll - x$empcov.maxpll
}
if(is.null(args$ylim)){
ylim <- c(max(-8, min(unlist(lapply(lik, min, na.rm = TRUE)))), 0)
} else{
ylim <- args$ylim
}
tikz <- FALSE
level <- c(0.95, 0.99)
if (!is.null(args$level)) {
level <- args$level[1]
}
if (!is.null(args$tikz)) {
if (isTRUE(args$tikz)) {
tikz <- TRUE
}
}
if (any(!is.null(args$which), !is.null(args$ind), !is.null(args$parm))) {
if (!is.null(args$parm)) {
parm <- match.arg(arg = args$parm, choices = c("profile", "tem", "modif.tem", "modif.empcov",
"r", "rstar"), several.ok = TRUE)
parm[parm %in% "r"] <- "profile"
parm[parm %in% "rstar"] <- "tem"
ind <- which(c("profile", "tem", "modif.tem", "modif.empcov") %in% parm)
} else if (!is.null(args$ind)) {
ind <- args$ind[args$ind %in% 1:4]
parm <- c("profile", "tem", "modif.tem", "modif.empcov")[ind]
} else if (!is.null(args$which)) {
ind <- args$which[args$which %in% 1:4]
parm <- c("profile", "tem", "modif.tem", "modif.empcov")[ind]
}
parm <- unique(parm)
ind <- unique(ind[ind %in% 1:4])
} else {
ind <- 1:4
parm <- c("profile", "tem", "modif.tem", "modif.empcov")
}
if(is.null(args$xlim)){
xlim <- c(min(x$psi), max(x$psi))
} else{
xlim <- args$xlim
}
plot(NULL, type = "n", bty = "l", xlim = xlim, ylim = ylim, xlab = ifelse(!tikz,
expression(psi), "$\\psi$"), ylab = "Profile log likelihood")
abline(h = -qchisq(level, 1)/2, col = "gray")
# Legend
lcols <- NULL
llty <- NULL
llwd <- NULL
llegend <- NULL
if (4 %in% ind && !is.null(x$empcov.mle)) {
abline(v = x$empcov.mle, lwd = 0.5, col = 4, lty = 4)
}
if (3 %in% ind && !is.null(x$tem.mle)) {
abline(v = x$tem.mle, lwd = 0.5, col = 2, lty = 2)
}
if (2 %in% ind && !is.null(x$tem.psimax)) {
abline(v = x$tem.psimax, lwd = 0.5, col = 3)
}
abline(v = x$psi.max, lwd = 0.5)
if (4 %in% ind && !is.null(lik$empcov.npll)) {
lines(x$psi, lik$empcov.npll, lty = 4, col = 4, lwd = 2)
lcols <- c(lcols, 4)
llty <- c(llty, 4)
llwd <- c(llwd, 2)
llegend <- c(llegend, "modif. emp. cov.")
}
if (3 %in% ind && !is.null(lik$tem.npll)) {
lines(x$psi, lik$tem.npll, lty = 2, col = 2, lwd = 2)
lcols <- c(lcols, 2)
llty <- c(llty, 2)
llwd <- c(llwd, 2)
llegend <- c(llegend, "modif. tem.")
}
if (2 %in% ind && !is.null(x$rstar)) {
lines(x$psi, -x$rstar^2/2, lwd = 2, col = 3, lty = 5)
lcols <- c(lcols, 3)
llty <- c(llty, 5)
llwd <- c(llwd, 2)
llegend <- c(llegend, "tem")
}
lcols <- c(lcols, 1)
llty <- c(llty, 1)
llwd <- c(llwd, 2)
llegend <- c(llegend, "profile")
lines(x$psi, lik$npll, lwd = 2)
# add the legend in the top right corner
if(!isTRUE(all.equal(llegend, "profile"))){
legend(x = "topright", legend = rev(llegend), lty = rev(llty), lwd = rev(llwd), col = rev(lcols),
bty = "n", x.intersp = 0.2, seg.len = 0.5, cex = 0.9)
}
#par(old.pars)
}
#' Profile log-likelihood for the generalized extreme value distribution
#'
#' This function calculates the profile likelihood along with two small-sample corrections
#' based on Severini's (1999) empirical covariance and the Fraser and Reid tangent exponential
#' model approximation.
#'
#' @details The two additional \code{mod} available are \code{tem}, the tangent exponential model (TEM) approximation and
#' \code{modif} for the penalized profile likelihood based on \eqn{p^*} approximation proposed by Severini.
#' For the latter, the penalization is based on the TEM or an empirical covariance adjustment term.
#'
#' @param psi parameter vector over which to profile (unidimensional)
#' @param param string indicating the parameter to profile over
#' @param mod string indicating the model, one of \code{profile}, \code{tem} or \code{modif}.See \bold{Details}.
#' @param dat sample vector
#' @param N size of block over which to take maxima. Required only for \code{param} \code{Nmean} and \code{Nquant}.
#' @param p tail probability. Required only for \code{param} \code{quant}.
#' @param q probability level of quantile. Required only for \code{param} \code{Nquant}.
#' @param correction logical indicating whether to use \code{spline.corr} to smooth the tem approximation.
#' @param plot logical; should the profile likelihood be displayed? Default to \code{TRUE}
#' @param ... additional arguments such as output from call to \code{Vfun} if \code{mode='tem'}.
#'
#' @return a list with components
#' \itemize{
#' \item{\code{mle}:} maximum likelihood estimate
#' \item{\code{psi.max}:} maximum profile likelihood estimate
#' \item{\code{param}:} string indicating the parameter to profile over
#' \item{\code{std.error}:} standard error of \code{psi.max}
#' \item{\code{psi}:} vector of parameter \eqn{psi} given in \code{psi}
#' \item{\code{pll}:} values of the profile log likelihood at \code{psi}
#' \item{\code{maxpll}:} value of maximum profile log likelihood
#' }
#'
#'
#' In addition, if \code{mod} includes \code{tem}
#' \itemize{
#' \item{\code{normal}:}{maximum likelihood estimate and standard error of the interest parameter \eqn{psi}}
#' \item{\code{r}:}{values of likelihood root corresponding to \eqn{\psi}}
#' \item{\code{q}:}{vector of likelihood modifications}
#' \item{\code{rstar}:}{modified likelihood root vector}
#' \item{\code{rstar.old}:}{uncorrected modified likelihood root vector}
#' \item{\code{tem.psimax}:}{maximum of the tangent exponential model likelihood}
#' }
#' In addition, if \code{mod} includes \code{modif}
#' \itemize{
#' \item{\code{tem.mle}:} maximum of tangent exponential modified profile log likelihood
#' \item{\code{tem.profll}:} values of the modified profile log likelihood at \code{psi}
#' \item{\code{tem.maxpll}:} value of maximum modified profile log likelihood
#' \item{\code{empcov.mle}:} maximum of Severini's empirical covariance modified profile log likelihood
#' \item{\code{empcov.profll}:} values of the modified profile log likelihood at \code{psi}
#' \item{\code{empcov.maxpll}:} value of maximum modified profile log likelihood
#' }
#'
#' @references Fraser, D. A. S., Reid, N. and Wu, J. (1999), A simple general formula for tail probabilities for frequentist and Bayesian inference. \emph{Biometrika}, \bold{86}(2), 249--264.
#' @references Severini, T. (2000) Likelihood Methods in Statistics. Oxford University Press. ISBN 9780198506508.
#' @references Brazzale, A. R., Davison, A. C. and Reid, N. (2007) Applied asymptotics: case studies in small-sample statistics. Cambridge University Press, Cambridge. ISBN 978-0-521-84703-2
#'
#' @export
#' @examples
#' \dontrun{
#' set.seed(123)
#' dat <- rgev(n = 100, loc = 0, scale = 2, shape = 0.3)
#' gev.pll(psi = seq(0,0.5, length = 50), param = 'shape', dat = dat)
#' gev.pll(psi = seq(-1.5, 1.5, length = 50), param = 'loc', dat = dat)
#' gev.pll(psi = seq(10, 40, length = 50), param = 'quant', dat = dat, p = 0.01)
#' gev.pll(psi = seq(12, 100, length = 50), param = 'Nmean', N = 100, dat = dat)
#' gev.pll(psi = seq(12, 90, length = 50), param = 'Nquant', N = 100, dat = dat, q = 0.5)
#' }
gev.pll <- function(psi, param = c("loc", "scale", "shape", "quant", "Nmean", "Nquant"),
mod = "profile", dat, N = NULL, p = NULL, q = NULL, correction = TRUE, plot = TRUE, ...) {
param <- match.arg(param)
mod <- match.arg(mod, c("profile","tem", "modif"), several.ok = TRUE)
# Parametrization profiling for quant over scale is more numerically stable
if (param == "quant") {
stopifnot(!is.null(p))
q <- 1 - p
N = 1
param <- "Nquant"
}
oldpar <- match.arg(param, choices = c("loc", "scale", "shape", "quant", "Nmean", "Nquant"), several.ok = FALSE)
# Arguments for parametrization of the log likelihood
if (param %in% c("loc", "scale", "shape")) {
args <- c("loc", "scale", "shape")
} else if (param == "quant") {
args <- c(param, "scale", "shape")
} else {
args <- c("loc", param, "shape")
}
# Sanity checks to ensure all arguments are provided
if (is.null(N)) {
if (param %in% c("Nmean", "Nquant")) {
stop("Argument \"N\" missing. Procedure aborted")
} else {
N <- NA
}
}
if (is.null(q)) {
if (param == "Nquant") {
stop("Argument \"q\" missing. Procedure aborted")
} else {
q <- NA
}
}
if (is.null(p)) {
if (param == "quant") {
stop("Argument \"p\" missing. Procedure aborted")
} else {
p <- NA
}
}
xmin <- min(dat)
xmax <- max(dat)
# Find maximum likelihood estimates
mle <- gev.mle(xdat = dat, args = args, q = q, N = N, p = p)
# if(missing(psi) || any(is.null(psi)) || any(is.na(psi))){ psi <- mle[param] } psi <- as.vector(psi)
# Extract the components, notably V for model `tem`. Keep other components for optimization
Vprovided <- FALSE
extra.args <- list(...)
if ("V" %in% names(extra.args)) {
V <- extra.args$V
extra.args$V <- NULL
if (isTRUE(all.equal(dim(V), c(length(dat), 2)))) {
Vprovided <- TRUE
}
}
if (!Vprovided) {
V <- switch(param,
loc = gev.Vfun(par = mle, dat = dat),
scale = gev.Vfun(par = mle, dat = dat),
shape = gev.Vfun(par = mle, dat = dat),
quant = gevr.Vfun(par = mle, dat = dat, p = p),
Nmean = gevN.Vfun(par = mle, dat = dat, N = N, qty = "mean"),
Nquant = gevN.Vfun(par = mle, dat = dat, q = q, N = N, qty = "quantile"))
}
# Obtained constrained maximum likelihood estimates for given value of psi
if (param %in% c("loc", "scale", "shape")) {
# Define observation-wise gradient
gev.score.f <- function(par, dat) {
dat <- as.vector(dat)
mu = par[1]
sigma = par[2]
xi = as.vector(par[3])
if (!isTRUE(all.equal(xi, 0, tolerance = 1e-8))) {
cbind(-(-(mu - dat) * xi/sigma + 1)^(-1/xi - 1)/sigma - xi * (1/xi + 1)/(sigma *
((mu - dat) * xi/sigma - 1)), -(dat - mu) * ((dat - mu) * xi/sigma + 1)^(-1/xi -
1)/sigma^2 + (dat - mu) * xi * (1/xi + 1)/(sigma^2 * ((dat - mu) * xi/sigma +
1)) - 1/sigma, -(mu - dat) * (1/xi + 1)/(sigma * ((mu - dat) * xi/sigma -
1)) - (log1p(-(mu - dat) * xi/sigma)/xi^2 - (mu - dat)/(sigma * ((mu -
dat) * xi/sigma - 1) * xi))/(-(mu - dat) * xi/sigma + 1)^(1/xi) + log1p(-(mu -
dat) * xi/sigma )/xi^2)
} else {
cbind(-exp(mu/sigma - dat/sigma)/sigma + 1/sigma, mu * exp(mu/sigma - dat/sigma)/sigma^2 -
dat * exp(mu/sigma - dat/sigma)/sigma^2 - mu/sigma^2 - 1/sigma + dat/sigma^2,
rep(0, length(dat)))
}
}
ind <- switch(param, loc = 1, scale = 2, shape = 3)
maxll <- gev.ll(mle, dat = dat)
std.error <- sqrt(solve(gev.infomat(par = mle, dat = dat, method = "exp"))[ind, ind])
constr.mle.scale <- function(sigmat, dat = dat) {
x0 = c(median(dat/sigmat), 0.05)
if (is.nan(gev.ll(c(x0[1], sigmat, x0[2]), dat = dat))) {
constr_fit <- try(mev::fit.gev(xdat = dat, fpar = list(scale = sigmat, shape = x0[2])))
if (!inherits(constr_fit, what = "try-error")) {
if (constr_fit$convergence == "successful") {
x0 <- as.vector(c(constr_fit$estimate["loc"], 0.05))
} else {
stop("Could not find starting values for optimization routine")
}
} else {
stop("Could not find starting values for optimization routine")
}
}
# opt <- suppressMessages(nloptr::sbplx(x0 = x0, fn = function(par) {
# -gev.ll(c(par[1], sigmat, par[2]), dat = dat)
# }))
# opt2 <- suppressMessages(nloptr::slsqp(x0 = opt$par, fn = function(par) {
# -gev.ll(c(par[1], sigmat, par[2]), dat = dat)
# }, gr = function(par) {
# -gev.score(c(par[1], sigmat, par[2]), dat = dat)[-2]
# }))
opt <- try(suppressWarnings(suppressMessages(alabama::auglag(par = x0, fn = function(par) {
-gev.ll(c(par[1], sigmat, par[2]), dat = dat)
}, gr = function(par) {
-gev.score(c(par[1], sigmat, par[2]), dat = dat)[-2]
},
hin = function(par) {
ifelse(par[2] <= 0, sigmat + par[2] * (xmax - par[1]), sigmat + par[2] * (xmin -
par[1]))
}, control.outer = list(trace = FALSE, method = "nlminb")))))
if (!inherits(opt, what = "try-error")) {
if (opt$convergence == 0 && !isTRUE(all.equal(opt$value, 1e+10))) {
return(c(opt$par, opt$value))
}
}
opt2 <- try(suppressWarnings(suppressMessages(Rsolnp::solnp(pars = x0, fun = function(par) {
-gev.ll(c(par[1], sigmat, par[2]), dat = dat)
}, control = list(trace = 0)))))
if (!inherits(opt2, what = "try-error")) {
if (opt2$convergence == 0 && !isTRUE(all.equal(tail(opt2$values,1), 1e+10))) {
return(c(opt2$par, tail(opt2$values,1)))
}
}
return(rep(NA, 3))
}
constr.mle.loc <- function(mut, dat = dat) {
opt <- try(suppressMessages(suppressWarnings(
alabama::auglag(par = c(mad(dat, constant = 1), 0.1), fn = function(par) {
val <- -gev.ll(c(mut, par[1:2]), dat = dat)
ifelse(is.infinite(val), 1e+10, val)
}, gr = function(par) {
-gev.score(c(mut, par[1:2]), dat = dat)[-ind]
}, hin = function(par) {
ifelse(par[2] <= 0, par[1] + par[2] * (xmax - mut), par[1] + par[2] * (xmin -
mut))
}, control.outer = list(method = "nlminb", trace = FALSE)))))
if (!inherits(opt, what = "try-error")) {
if (opt$convergence == 0 && !isTRUE(all.equal(opt$value, 1e+10))) {
return(c(opt$par, opt$value))
}
}
opt2 <- try(suppressWarnings(suppressMessages(
Rsolnp::solnp(pars = c(mad(dat, constant = 1), 0.1),
fun = function(par) {
-gev.ll(c(mut, par), dat = dat)
}, control = list(trace = 0)))))
if (!inherits(opt2, what = "try-error")) {
if (opt2$convergence == 0 && !isTRUE(all.equal(tail(opt2$values,1), 1e+10))) {
return(c(opt2$par, tail(opt2$values,1)))
}
}
return(rep(NA, 3))
}
# constr.mle.shape <- function(xit, dat = dat){
# start.scale <- max(1e-2,mad(dat, constant = 1)/median(abs(mev::rgev(10000, shape=xit)-mev::qgev(0.5, shape=xit))))
# start.loc <- median(dat) - start.scale*ifelse(xit == 0, log(log(2)), (log(2)^(-xit)-1)/xit)
# if(any(c(start.scale + xit*(xmax-start.loc) <= 0, start.scale + xit*(xmin-start.loc) <= 0)))
# {
# if(xit < 0){ start.loc <- start.scale/xit + xmax + 1e-3
# } else { start.loc <- start.scale/xit + xmin + 1e-3 } }
# #Subplex - simplex type algorithm, more robust than Nelder-Mead
# opt <- nloptr::sbplx(x0 = c(start.loc, start.scale),
# fn = function(par){-gev.ll(c(par,xit), dat=dat)}, lower= c(-Inf, 0),
# control = list(xtol_rel = 1e-8, maxeval = 2000, ftol_rel = 1e-10))
# #If solution is not within the region
# if(ifelse(xit < 0, opt$par[2] + xit*(xmax-opt$par[1]) <= 0, opt$par[2] + xit*(xmin-opt$par[1]) <= 0))
# {
# opt <- nloptr::auglag(x0 = c(start.loc, start.scale), fn = function(par){
# val <- gev.ll.optim(c(par[1:2], xit), dat = dat);
# ifelse(is.infinite(val) || is.na(val), 1e10, val)},
# gr = function(par){ val <- -gev.score(c(par[1], exp(par[2]), xit), dat = dat)[-ind]},
# hin = function(par){c(ifelse(xit <= 0, exp(par[2]) + xit*(xmax-par[1]), exp(par[2]) + xit*(xmin-par[1])))} ) }
# if(!inherits(opt, what = "try-error")){
# if(all(c(opt$convergence > 0, abs(gev.score(c(opt$par, xit), dat = dat)[1:2]) < 5e-4))){
# return(c(opt$par, opt$value)) } else { #mev::fgev(start = list(loc = opt$par[1], scale =
# opt$par[2]), shape = xit, # x = dat, method = 'BFGS', control=list(reltol=1e-10, abstol =
# 1e-9)) opt2 <- suppressWarnings(nloptr::slsqp(x0 = opt$par, fn = function(par){ val <-
# gev.ll.optim(c(par[1:2], xit), dat = dat); ifelse(is.infinite(val) || is.na(val), 1e10,
# val)}, gr = function(par){ val <- -gev.score(c(par[1], exp(par[2]), xit), dat =
# dat)[-ind]}, hin = function(par){c(ifelse(xit <= 0, exp(par[2]) + xit*(xmax-par[1]),
# exp(par[2]) + xit*(xmin-par[1])))} )) opt2$par[2] <- exp(opt2$par[2])
# if(all(c(opt2$convergence > 0, !isTRUE(all.equal(opt2$value, 1e10)),
# abs(gev.score(c(opt2$par, xit), dat = dat)[1:2]) < 1e-4))){ return(c(opt2$par,
# opt2$value)) } } } return(rep(NA, 3)) }
constr.mle.shape <- function(xit, dat = dat) {
if (abs(xit) < 1e-08) {
xit <- 0
} #because rgev does not handle this case!
start.scale <- mad(dat, constant = 1)/median(abs(mev::rgev(2000, shape = xit) -
mev::qgev(0.5, shape = xit)))
start.loc <- median(dat) - start.scale * ifelse(xit == 0, log(log(2)), (log(2)^(-xit) -
1)/xit)
if (start.scale + xit * (xmax - start.loc) <= 0) {
if (xit < 0) {
start.loc <- start.scale/xit + xmax + 0.001
} else {
start.loc <- start.scale/xit + xmin + 0.001
}
}
opt <- try(suppressMessages(suppressWarnings(
alabama::auglag(par = c(start.loc, start.scale),
fn = function(par) {
val <- -gev.ll(c(par[1:2], xit), dat = dat)
ifelse(is.infinite(val) || is.na(val), 1e+10, val)
}, gr = function(par) {
-gev.score(c(par[1:2], xit), dat = dat)[-ind]
},
hin = function(par) {
ifelse(xit <= 0, par[2] + xit * (xmax - par[1]), par[2] + xit * (xmin - par[1]))
}, control.outer = list(method = "nlminb", trace = FALSE)))))
if (!inherits(opt, what = "try-error")) {
if (opt$convergence == 0 && !isTRUE(all.equal(opt$value, 1e+10))) {
return(c(opt$par, opt$value))
}
}
opt2 <- try(suppressMessages(suppressWarnings(
Rsolnp::solnp(pars = c(start.loc, start.scale),
fun = function(par) {
val <- -gev.ll(c(par[1:2], xit), dat = dat)
ifelse(is.infinite(val) || is.na(val), 1e+10, val)
}, ineqfun = function(par) {
ifelse(xit <= 0, par[2] + xit * (xmax - par[1]), par[2] + xit * (xmin -
par[1]))
}, ineqLB = 0, ineqUB = Inf, control = list(trace = 0)))))
if (!inherits(opt2, what = "try-error")) {
if (isTRUE(opt2$convergence == 0) && !isTRUE(all.equal(tail(opt2$values,1), 1e+10))) {
return(c(opt2$par, tail(opt2$values,1)))
}
}
return(rep(NA, 3))
}
# Missing psi vector
if (missing(psi) || any(is.null(psi)) || any(is.na(psi))) {
if (ind == 2) {
psirangelow <- unique(min(1e-05, seq(-4, -1.5, length = 10) * std.error + mle[param]))
} else {
psirangelow <- seq(-4, -1.5, length = 10) * std.error + mle[param]
}
lowvals <- -t(sapply(psirangelow, function(par) {
switch(param, loc = constr.mle.loc(par, dat = dat), scale = constr.mle.scale(par,
dat = dat), shape = constr.mle.shape(par, dat = dat))
}))[, 3] - maxll
psirangehigh <- seq(1.5, 4, length = 10) * std.error + mle[param]
highvals <- -t(sapply(psirangehigh, function(par) {
switch(param, loc = constr.mle.loc(par, dat = dat), scale = constr.mle.scale(par,
dat = dat), shape = constr.mle.shape(par, dat = dat))
}))[, 3] - maxll
lo <- approx(x = lowvals, y = psirangelow, xout = -4)$y
hi <- approx(x = highvals, y = psirangehigh, xout = -4)$y
lo <- ifelse(is.na(lo), predict(lm(psirangelow ~ lowvals), newdata = data.frame(lowvals = -4))[[1]],
lo)
hi <- ifelse(is.na(hi), predict(lm(psirangehigh ~ highvals), newdata = data.frame(highvals = -4))[[1]],
hi)
psi <- seq(lo, hi, length = 55)
}
if (param == "shape" && min(psi) < -1) {
warning("psi includes shape parameters below -1. These will be automatically removed.")
psi <- psi[psi > -1]
}
# Calculate profile likelihood at psi
lambda <- t(sapply(psi, function(par) {
switch(param, loc = constr.mle.loc(par, dat = dat), scale = constr.mle.scale(par,
dat = dat), shape = constr.mle.shape(par, dat = dat))
}))
pars <- switch(param, loc = cbind(psi, lambda[, 1:2, drop = FALSE]), scale = cbind(lambda[,
1, drop = FALSE], psi, lambda[, 2, drop = FALSE]), shape = cbind(lambda[, 1:2,
drop = FALSE], psi))
# Profile log likelihood values for psi
profll <- -lambda[, 3]
r <- sign(mle[param] - psi) * sqrt(2 * (maxll - profll))
if ("tem" %in% mod) {
# Tangent exponential model approximation of Fraser and Reid to the profile likelihood
phi.mle <- gev.phi(par = mle, dat = dat, V = V)
q2num <- ifelse(ind%%2 == 0, -1, 1) * apply(pars, 1, function(par) {
det(rbind(c(c(phi.mle) - gev.phi(par = par, dat = dat, V = V)), gev.dphi(par = par,
dat = dat, V = V)[-ind, ]))
})
if (isTRUE(any(sign(q2num) * sign(r) < 0, na.rm = TRUE))) {
warning("Correction factor and likelihood root are of opposite sign - check output")
}
logq <- apply(pars, 1, function(par) {
-0.5 * log(det(gev.infomat(par = par, dat = dat, method = "obs")[-ind, -ind]))
}) + log(abs(q2num))
qmlecontrib <- -log(det(gev.dphi(par = mle, dat = dat, V = V))) + 0.5 * log(det(gev.infomat(par = mle,
dat = dat, method = "obs")))
logq <- logq + qmlecontrib
qcor <- sign(q2num) * exp(logq)
rstar <- ifelse(r == 0, 0, r + (logq - log(abs(r)))/r)
###
tem.max.opt <- function(psi, dat = dat) {
lambda <- switch(param, loc = constr.mle.loc(psi, dat = dat), scale = constr.mle.scale(psi,
dat = dat), shape = constr.mle.shape(psi, dat = dat))[1:2]
para <- switch(ind, c(psi, lambda), c(lambda[1], psi, lambda[2]), c(lambda,
psi))
pll <- gev.ll(par = para, dat = dat)
rs <- 2 * (maxll - pll)
logq <- -0.5 * log(det(gev.infomat(par = para, dat = dat, method = "obs")[-ind,
-ind])) + qmlecontrib + log(abs(det(rbind(c(c(phi.mle) - gev.phi(par = para,
dat = dat, V = V)), gev.dphi(par = para, dat = dat, V = V)[-ind, ]))))
abs(rs + logq - log(sqrt(abs(rs))))
# rs is r squared - finding the maximum by multiplying rstar by r
# when rstar vanishes at zero, so does rstar*r
# this ensures that we do not divide by r = zero
}
tem.max <- optim(par = mle[ind], fn = tem.max.opt, method = "Brent", dat = dat,
lower = ifelse(ind == 2, max(1e-05, mle[ind] - std.error), mle[ind] - std.error),
upper = mle[ind] + std.error, control = list(abstol = 1e-10))$par
###
}
# Modified profile likelihood based on p* approximation, two modifications due to Severini
if ("modif" %in% mod) {
tem.objfunc <- function(par, dat = dat) {
0.5 * log(det(gev.infomat(par, dat = dat, method = "obs")[-ind, -ind])) -
log(abs(det(gev.dphi(par = par, dat = dat, V = V[, -ind])[-ind, ])))
}
optim.tem.fn <- function(psi, dat = dat, param = param) {
theta.psi.opt <- switch(param, loc = constr.mle.loc(psi, dat = dat), scale = constr.mle.scale(psi,
dat = dat), shape = constr.mle.shape(psi, dat = dat))
para <- switch(param, loc = c(psi, theta.psi.opt[1:2]), scale = c(theta.psi.opt[1],
psi, theta.psi.opt[2]), shape = c(theta.psi.opt[1:2], psi))
ll <- -theta.psi.opt[3]
ll + tem.objfunc(para, dat = dat)
}
# TEM profile log likelihood values for psi
proflltem <- profll + apply(pars, 1, tem.objfunc, dat = dat)
# Maximum objective function for TEM (line search in neighborhood of the MLE)
tem.mle.opt <- optim(par = mle[ind], fn = optim.tem.fn, method = "Brent", dat = dat,
param = param, lower = ifelse(param == "scale", max(1e-05, mle[ind] - std.error),
mle[ind] - std.error), upper = mle[ind] + std.error, control = list(fnscale = -1))
# Severini empirical covariance function adjustment to profile likelihood Score function -
# observation-wise
# Score at MLE (sums to zero)
score.scale.mle <- gev.score.f(mle, dat)[, -ind] #keep s_lambda
empcov.objfunc <- function(par, dat) {
0.5 * log(det(gev.infomat(par = par, dat = dat, method = "obs")[-ind, -ind])) -
log(abs(sum(score.scale.mle * gev.score.f(par, dat)[, -ind])))
}
profllempcov <- profll + apply(pars, 1, empcov.objfunc, dat = dat)
optim.empcov.fn <- function(psi, param = param, dat = dat) {
theta.psi.opt <- switch(param, loc = constr.mle.loc(psi, dat = dat), scale = constr.mle.scale(psi,
dat = dat), shape = constr.mle.shape(psi, dat = dat))
para <- switch(param, loc = c(psi, theta.psi.opt[1:2]), scale = c(theta.psi.opt[1],
psi, theta.psi.opt[2]), shape = c(theta.psi.opt[1:2], psi))
ll <- -theta.psi.opt[3]
ll + empcov.objfunc(para, dat = dat)
}
empcov.mle.opt <- optim(par = mle[ind], fn = optim.empcov.fn, method = "Brent",
dat = dat, param = param, lower = ifelse(param == "scale", max(1e-05, mle[ind] -
std.error), mle[ind] - std.error), upper = mle[ind] + std.error, control = list(fnscale = -1))
}
# Return levels, quantiles or value-at-risk
} else if (param %in% c("Nquant", "Nmean")) {
qty <- switch(param, Nquant = "quantile", Nmean = "mean")
maxll <- gevN.ll(mle, dat = dat, N = N, q = q, qty = qty)
std.error <- sqrt(solve(gevN.infomat(par = mle, dat = dat, method = "exp", N = N, q = q,
qty = qty))[2, 2])
constr.mle.N <- function(zt, dat = dat) {
st_vals <- c(median(dat), 0.1)
if (isTRUE(as.vector(mle["shape"] > 0))) {
st_vals <- mle[c("loc", "shape")]
}
# COBYLA unfortunately hangs from time to time. so it cannot be used in optimization
# routine without risking hanging despite the algorithm begin more robust and faster for
# this problem...
opt <- try(suppressMessages(suppressWarnings(alabama::auglag(par = st_vals, fn = function(par) {
val <- -gevN.ll(par = c(par[1], zt, par[2]), dat = dat, q = q, qty = qty, N = N)
ifelse(isTRUE(any(is.infinite(val), is.na(val), val <= -maxll)), 1e+10, val)
}, hin = function(par) {
sigma <- switch(qty, quantile = (zt - par[1]) * par[2]/(N^par[2] * (log(1/q))^(-par[2]) -
1), mean = (zt - par[1]) * par[2]/(N^par[2] * gamma(1 - par[2]) - 1))
c(sigma, sigma + par[2] * (xmax - par[1]), sigma + par[2] * (xmin - par[1]))
}, control.outer = list(method = "nlminb", trace = FALSE)))))
#With `alabama` package opt <- try(suppressWarnings( alabama::auglag(par = c(median(dat),
# 0.1), fn = function(par){ val <- -gevN.ll(par = c(par[1], zt, par[2]), dat = dat, q = q,
# qty = qty, N = N); ifelse(is.infinite(val) || is.na(val), 1e10, val)}, gr =
# function(par){-gevN.score(par = c(par[1], zt, par[2]), dat = dat, q = q, qty = qty, N =
# N)[-2]}, hin = function(par){ sigma <- switch(qty, quantile =
# (zt-par[1])*par[2]/(N^par[2]*(log(1/q))^(-par[2])-1), mean =
# (zt-par[1])*par[2]/(N^par[2]*gamma(1-par[2])-1)) c(sigma, sigma + par[2]*(xmax-par[1]),
# sigma + par[2]*(xmin-par[1]))}, control.outer = list (trace = FALSE) )))
if (!inherits(opt, what = "try-error")) {
# if(opt$convergence == 0 && !isTRUE(all.equal(opt$value, 1e10))){
if (isTRUE(opt$convergence == 0) && !isTRUE(all.equal(opt$value, 1e+10))) {
return(c(opt$par, opt$value))
} else{
st_vals <- opt$par
}
}
opt2 <- try(suppressMessages(suppressWarnings(
Rsolnp::solnp(par = st_vals, fun = function(par) {
val <- -gevN.ll(par = c(par[1], zt, par[2]), dat = dat, q = q, qty = qty,
N = N)
ifelse(is.infinite(val) || is.na(val), 1e+10, val)
}, ineqfun = function(par) {
sigma <- switch(qty, quantile = (zt - par[1]) * par[2]/(N^par[2] * (log(1/q))^(-par[2]) -
1), mean = (zt - par[1]) * par[2]/(N^par[2] * gamma(1 - par[2]) - 1))
c(sigma, sigma + par[2] * (xmax - par[1]), sigma + par[2] * (xmin - par[1]))
}, ineqLB = rep(0, 3), ineqUB = rep(Inf, 3),
control = list(trace = 0)))))
if (!inherits(opt2, what = "try-error")) {
if (isTRUE(opt2$convergence == 0) && !isTRUE(all.equal(tail(opt2$values,1), 1e+10))) {
return(c(opt2$par, tail(opt2$values,1)))
}
}
return(rep(NA,3))
}
# Missing psi vector
if (missing(psi) || any(is.null(psi)) || any(is.na(psi))) {
psirangelow <- pmax(1e-05, seq(ifelse(mle[3] < 0, -3.5, -2.5), -0.5, length = 6) *
std.error + mle[2])
lowvals <- -sapply(psirangelow, constr.mle.N, dat = dat)[3, ] - maxll
psirangehigh <- seq(2.5, (1 + mle[3]) * 8, length = 10) * std.error + mle[2]
highvals <- -sapply(psirangehigh, constr.mle.N, dat = dat)[3, ] - maxll
lo <- approx(x = lowvals, y = psirangelow, xout = -4)$y
hi <- approx(x = highvals, y = psirangehigh, xout = -4)$y
lo <- ifelse(is.na(lo), lm(psirangelow ~ lowvals)$coef[2] * -4 + mle[2], lo)
hi <- ifelse(is.na(hi), lm(psirangehigh ~ highvals)$coef[2] * -4 + mle[2], hi)
psi <- c(seq(lo, mle[2], length = 20)[-20], seq(mle[2], hi, length = 55)[-1])
}
lambda <- t(sapply(psi, constr.mle.N, dat = dat))
pars <- cbind(lambda[, 1, drop = FALSE], psi, lambda[, 2, drop = FALSE])
# Profile log likelihood values for psi
profll <- -lambda[, 3]
r <- sign(mle[param] - psi) * sqrt(2 * (maxll - profll))
if ("tem" %in% mod) {
phi.mle <- gevN.phi(par = mle, dat = dat, V = V, N = N, q = q, qty = qty)
q2num <- apply(pars, 1, function(par) {
-det(rbind(c(c(phi.mle) - gevN.phi(par = par, dat = dat, V = V, N = N, q = q,
qty = qty)), gevN.dphi(par = par, dat = dat, V = V, N = N, q = q, qty = qty)[-2,
]))
})
if (isTRUE(any(sign(q2num) * sign(r) < 0, na.rm = TRUE))) {
warning("Correction factor and likelihood root are of opposite sign - check output")
}
logq <- apply(pars, 1, function(par) {
-0.5 * log(det(gevN.infomat(par = par, dat = dat, method = "obs", N = N, q = q,
qty = qty)[-2, -2]))
}) + log(abs(q2num))
qmlecontrib <- -log(det(gevN.dphi(par = mle, dat = dat, V = V, N = N, q = q, qty = qty))) +
0.5 * log(det(gevN.infomat(par = mle, dat = dat, method = "obs", N = N, q = q,
qty = qty)))
logq <- logq + qmlecontrib
qcor <- sign(q2num) * exp(logq)
rstar <- ifelse(r == 0, 0, r + (logq - log(abs(r)))/r)
tem.max.opt <- function(psi, dat = dat) {
lam <- constr.mle.N(psi, dat = dat)
para <- c(lam[1], psi, lam[2])
pll <- -lam[3]
rs <- 2 * (maxll - pll)
logq <- -0.5 * log(det(gevN.infomat(par = para, dat = dat, method = "obs",
N = N, q = q, qty = qty)[-2, -2])) + qmlecontrib + log(abs(det(rbind(c(c(phi.mle) -
gevN.phi(par = para, dat = dat, V = V, q = q, N = N, qty = qty)), gevN.dphi(par = para,
dat = dat, V = V, q = q, N = N, qty = qty)[-2, ]))))
abs(rs + logq - log(sqrt(abs(rs))))
}
tem.max <- optim(par = mle[2], fn = tem.max.opt, method = "Brent", dat = dat, lower = max(1e-05,
mle[2] - std.error), upper = mle[2] + std.error, control = list(abstol = 1e-10))$par
names(mle)[2] <- oldpar
}
if ("modif" %in% mod) {
# Tangent exponential model approximation of Fraser and Reid to the profile likelihood
tem.objfunc.N <- function(par, dat = dat, N = N, qty = qty, q = q) {
0.5 * log(det(gevN.infomat(par = par, dat = dat, method = "obs", N = N, qty = qty,
q = q)[-2, -2])) - log(abs(det(gevN.dphi(par = par, dat = dat, N = N, V = V[,
-2], qty = qty, q = q)[-2, ])))
}
optim.tem.fn.N <- function(psi, dat = dat, q = q, qty = qty, N = N) {
theta.psi.opt <- constr.mle.N(psi, dat = dat)
param <- c(theta.psi.opt[1], psi, theta.psi.opt[2])
ll <- gevN.ll(param, dat = dat, N = N, q = q, qty = qty)
ll + tem.objfunc.N(param, dat = dat, N = N, qty = qty, q = q)
}
# TEM profile log likelihood values for psi
proflltem <- profll + suppressWarnings(apply(pars, 1, tem.objfunc.N, dat = dat,
N = N, qty = qty, q = q))
# Maximum objective function for TEM
tem.mle.opt <- optim(par = mle[2], fn = optim.tem.fn.N, method = "Brent", dat = dat,
q = q, qty = qty, N = N, lower = max(min(dat), mle[2] - std.error), upper = mle[2] +
std.error, control = list(fnscale = -1))
# Severini empirical covariance function adjustment to profile likelihood Score function -
# observation-wise for xi
gevN.score.f <- function(par, dat, N, q = q, qty = qty) {
qty <- match.arg(qty, c("mean", "quantile"))
mu = par[1]
z = par[2]
xi = par[3]
Npxi <- N^xi
log1q <- log(1/q)
logN <- log(N)
if (qty == "quantile") {
# quantiles at prob. q
cbind(((-(Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu - z) + 1)^(-1/xi -
1) * ((Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu - z)^2 + (Npxi * log1q^(-xi) -
1)/(mu - z))/xi + ((Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu - z)^2 +
(Npxi * log1q^(-xi) - 1)/(mu - z)) * (1/xi + 1)/((Npxi * log1q^(-xi) -
1) * (dat - mu)/(mu - z) - 1) - 1/(mu - z)), (-(Npxi * log1q^(-xi) -
1) * (dat - mu) * (-(Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu - z) +
1)^(-1/xi - 1)/((mu - z)^2 * xi) - (Npxi * log1q^(-xi) - 1) * (dat -
mu) * (1/xi + 1)/(((Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu - z) -
1) * (mu - z)^2) + 1/(mu - z)), ((-(Npxi * log1q^(-xi) - 1) * (dat -
mu)/(mu - z) + 1)^(-1/xi) * ((Npxi * log1q^(-xi) * logN - Npxi * log1q^(-xi) *
log(log1q)) * (dat - mu)/(((Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu -
z) - 1) * (mu - z) * xi) - log1p(-(Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu -
z))/xi^2) - (Npxi * log1q^(-xi) * logN - Npxi * log1q^(-xi) *
log(log1q)) * (dat - mu) * (1/xi + 1)/(((Npxi * log1q^(-xi) - 1) *
(dat - mu)/(mu - z) - 1) * (mu - z)) + (Npxi * log1q^(-xi) - 1) * ((Npxi *
log1q^(-xi) * logN - Npxi * log1q^(-xi) * log(log1q)) * (mu -
z) * xi/(Npxi * log1q^(-xi) - 1)^2 - (mu - z)/(Npxi * log1q^(-xi) -
1))/((mu - z) * xi) + log1p(-(Npxi * log1q^(-xi) - 1) * (dat - mu)/(mu -
z))/xi^2))
} else {
# Mean
cbind(((-(Npxi * gamma(-xi + 1) - 1) * (dat - mu)/(mu - z) + 1)^(-1/xi -
1) * ((Npxi * gamma(-xi + 1) - 1) * (dat - mu)/(mu - z)^2 + (Npxi * gamma(-xi +
1) - 1)/(mu - z))/xi + ((Npxi * gamma(-xi + 1) - 1) * (dat - mu)/(mu -
z)^2 + (Npxi * gamma(-xi + 1) - 1)/(mu - z)) * (1/xi + 1)/((Npxi * gamma(-xi +
1) - 1) * (dat - mu)/(mu - z) - 1) - 1/(mu - z)), (-(Npxi * gamma(-xi +
1) - 1) * (dat - mu) * (-(Npxi * gamma(-xi + 1) - 1) * (dat - mu)/(mu -
z) + 1)^(-1/xi - 1)/((mu - z)^2 * xi) - (Npxi * gamma(-xi + 1) - 1) * (dat -
mu) * (1/xi + 1)/(((Npxi * gamma(-xi + 1) - 1) * (dat - mu)/(mu - z) -
1) * (mu - z)^2) + 1/(mu - z)), ((-(Npxi * gamma(-xi + 1) - 1) * (dat -
mu)/(mu - z) + 1)^(-1/xi) * ((Npxi * logN * gamma(-xi + 1) - Npxi * psigamma(-xi +
1) * gamma(-xi + 1)) * (dat - mu)/(((Npxi * gamma(-xi + 1) - 1) * (dat -
mu)/(mu - z) - 1) * (mu - z) * xi) - log1p(-(Npxi * gamma(-xi + 1) - 1) *
(dat - mu)/(mu - z))/xi^2) - (Npxi * logN * gamma(-xi + 1) - Npxi *
psigamma(-xi + 1) * gamma(-xi + 1)) * (dat - mu) * (1/xi + 1)/(((Npxi *
gamma(-xi + 1) - 1) * (dat - mu)/(mu - z) - 1) * (mu - z)) + (Npxi * gamma(-xi +
1) - 1) * ((Npxi * logN * gamma(-xi + 1) - Npxi * psigamma(-xi + 1) *
gamma(-xi + 1)) * (mu - z) * xi/(Npxi * gamma(-xi + 1) - 1)^2 - (mu - z)/(Npxi *
gamma(-xi + 1) - 1))/((mu - z) * xi) + log1p(-(Npxi * gamma(-xi + 1) - 1) *
(dat - mu)/(mu - z))/xi^2))
}
}
# Score at MLE (sums to zero)
score.N.mle <- gevN.score.f(mle, dat, N, qty = qty, q = q)
empcov.objfunc.N <- function(par, dat = dat, q = q, qty = qty, N = N) {
0.5 * log(det(gevN.infomat(par = par, dat = dat, method = "obs", N = N, q = q,
qty = qty)[-2, -2])) - log(abs(sum(score.N.mle * gevN.score.f(par, dat, N = N,
qty = qty, q = q))))
}
profllempcov <- profll + suppressWarnings(apply(pars, 1, empcov.objfunc.N, N = N,
q = q, dat = dat, qty = qty))
optim.empcov.fn.N <- function(psi, dat = dat, q = q, qty = qty, N = N) {
theta.psi.opt <- constr.mle.N(psi, dat = dat)
param <- c(theta.psi.opt[1], psi, theta.psi.opt[2])
ll <- gevN.ll(param, dat = dat, N = N, q = q, qty = qty)
ll + empcov.objfunc.N(param, dat = dat, q = q, qty = qty, N = N)
}
empcov.mle.opt <- optim(par = mle[2], fn = optim.empcov.fn.N, method = "Brent",
dat = dat, qty = qty, q = q, N = N, lower = max(min(dat), mle[2] - std.error),
upper = mle[2] + std.error, control = list(fnscale = -1))
}
}
# Return profile likelihood and quantities of interest (modified likelihoods)
colnames(pars) <- names(mle)
ans <- list(mle = mle, pars = pars, psi.max = as.vector(mle[oldpar]), param = oldpar, std.error = std.error,
psi = psi, pll = profll, maxpll = maxll, r = r)
if ("tem" %in% mod) {
ans$q <- qcor
ans$rstar <- rstar
ans$normal <- c(ans$psi.max, ans$std.error)
if (correction && length(psi) > 10) {
ans <- spline.corr(ans)
}
ans$tem.psimax <- tem.max
}
if ("modif" %in% mod) {
ans$tem.mle <- tem.mle.opt$par
ans$tem.pll <- proflltem
ans$tem.maxpll <- tem.mle.opt$value
ans$empcov.mle <- empcov.mle.opt$par
ans$empcov.pll <- profllempcov
ans$empcov.maxpll <- empcov.mle.opt$value
}
if ("tem" %in% mod) {
class(ans) <- c("eprof", "fr")
} else {
class(ans) <- "eprof"
}
ans$family <- "gev"
if(plot){
plot(ans)
}
return(invisible(ans))
}
#' Profile log-likelihood for the generalized Pareto distribution
#'
#' This function calculates the (modified) profile likelihood based on the \eqn{p^*} formula.
#' There are two small-sample corrections that use a proxy for
#' \eqn{\ell_{\lambda; \hat{\lambda}}}{the sample space derivative of the nuisance},
#' which are based on Severini's (1999) empirical covariance
#' and the Fraser and Reid tangent exponential model approximation.
#' @details The three \code{mod} available are \code{profile} (the default), \code{tem}, the tangent exponential model (TEM) approximation and
#' \code{modif} for the penalized profile likelihood based on \eqn{p^*} approximation proposed by Severini.
#' For the latter, the penalization is based on the TEM or an empirical covariance adjustment term.
#'
#' @param psi parameter vector over which to profile (unidimensional)
#' @param param string indicating the parameter to profile over
#' @param mod string indicating the model. See \bold{Details}.
#' @param mle maximum likelihood estimate in \eqn{(\psi, \xi)} parametrization if \eqn{\psi \neq \xi} and \eqn{(\sigma, \xi)} otherwise (optional).
#' @param dat sample vector of excesses, unless \code{threshold} is provided (in which case user provides original data)
#' @param m number of observations of interest for return levels. Required only for \code{args} values \code{'VaR'} or \code{'ES'}
#' @param N size of block over which to take maxima. Required only for \code{args} \code{Nmean} and \code{Nquant}.
#' @param p tail probability, equivalent to \eqn{1/m}. Required only for \code{args} \code{quant}.
#' @param q level of quantile for N-block maxima. Required only for \code{args} \code{Nquant}.
#' @param correction logical indicating whether to use \code{spline.corr} to smooth the tem approximation.
#' @param threshold numerical threshold above which to fit the generalized Pareto distribution
#' @param plot logical; should the profile likelihood be displayed? Default to \code{TRUE}
#' @param ... additional arguments such as output from call to \code{Vfun} if \code{mode='tem'}.
#'
#' @return a list with components
#' \itemize{
#' \item{\code{mle}:} maximum likelihood estimate
#' \item{\code{psi.max}:} maximum profile likelihood estimate
#' \item{\code{param}:} string indicating the parameter to profile over
#' \item{\code{std.error}:} standard error of \code{psi.max}
#' \item{\code{psi}:} vector of parameter \eqn{psi} given in \code{psi}
#' \item{\code{pll}:} values of the profile log likelihood at \code{psi}
#' \item{\code{maxpll}:} value of maximum profile log likelihood
#' \item{\code{family}:} a string indicating "gpd"
#' \item{\code{threshold}:} value of the threshold, by default zero
#' }
#' In addition, if \code{mod} includes \code{tem}
#' \itemize{
#' \item{\code{normal}:}{maximum likelihood estimate and standard error of the interest parameter \eqn{psi}}
#' \item{\code{r}:}{values of likelihood root corresponding to \eqn{\psi}}
#' \item{\code{q}:}{vector of likelihood modifications}
#' \item{\code{rstar}:}{modified likelihood root vector}
#' \item{\code{rstar.old}:}{uncorrected modified likelihood root vector}
#' \item{\code{tem.psimax}:}{maximum of the tangent exponential model likelihood}
#' }
#' In addition, if \code{mod} includes \code{modif}
#' \itemize{
#' \item{\code{tem.mle}:} maximum of tangent exponential modified profile log likelihood
#' \item{\code{tem.profll}:} values of the modified profile log likelihood at \code{psi}
#' \item{\code{tem.maxpll}:} value of maximum modified profile log likelihood
#' \item{\code{empcov.mle}:} maximum of Severini's empirical covariance modified profile log likelihood
#' \item{\code{empcov.profll}:} values of the modified profile log likelihood at \code{psi}
#' \item{\code{empcov.maxpll}:} value of maximum modified profile log likelihood
#' }
#' @export
#' @examples
#' \dontrun{
#' dat <- rgp(n = 100, scale = 2, shape = 0.3)
#' gpd.pll(psi = seq(-0.5, 1, by=0.01), param = 'shape', dat = dat)
#' gpd.pll(psi = seq(0.1, 5, by=0.1), param = 'scale', dat = dat)
#' gpd.pll(psi = seq(20, 35, by=0.1), param = 'quant', dat = dat, p = 0.01)
#' gpd.pll(psi = seq(20, 80, by=0.1), param = 'ES', dat = dat, m = 100)
#' gpd.pll(psi = seq(15, 100, by=1), param = 'Nmean', N = 100, dat = dat)
#' gpd.pll(psi = seq(15, 90, by=1), param = 'Nquant', N = 100, dat = dat, q = 0.5)
#' }
gpd.pll <- function(psi, param = c("scale", "shape", "quant", "VaR", "ES", "Nmean", "Nquant"),
mod = "profile", mle = NULL, dat, m = NULL, N = NULL, p = NULL, q = NULL, correction = TRUE, threshold = NULL, plot = TRUE,
...) {
param <- match.arg(param)
mod <- match.arg(mod, c("profile", "tem", "modif"), several.ok = TRUE)
#If there is a threshold
if(!is.null(threshold)){
stopifnot(is.numeric(threshold), length(threshold) == 1)
if(min(dat) < threshold){
dat <- dat[dat>threshold] - threshold
} else {
dat <- dat - threshold
}
} else {
threshold <- 0
}
# Arguments for parametrization of the log likelihood
if (param == "shape") {
args <- c("scale", "shape")
} else {
args <- c(param, "shape")
}
# Sanity checks to ensure all arguments are provided
if (is.null(N)) {
if (param %in% c("Nmean", "Nquant")) {
stop("Argument \"N\" missing. Procedure aborted")
} else {
N <- NA
}
}
if (is.null(m)) {
if (param %in% c("VaR", "ES")) {
stop("Argument \"m\" missing. Procedure aborted")
} else {
m <- NA
}
}
if (is.null(p)) {
if (param == "quant") {
stop("Argument \"p\" missing. Procedure aborted")
} else {
p <- NA
}
}
if (is.null(q)) {
if (param == "Nquant") {
stop("Argument \"q\" missing. Procedure aborted")
} else {
q <- NA
}
}
xmin <- min(dat)
xmax <- max(dat)
shiftres <- param %in% c("Nmean", "Nquant", "VaR", "quant")
# If maximum likelihood estimates are not provided, find them
if (is.null(mle) || length(mle) != 2) {
mle <- gpd.mle(xdat = dat, args = args, m = m, N = N, p = p, q = q)
}
# Extract the components, notably V for model `tem`. Keep other components for optimization
Vprovided <- FALSE
extra.args <- list(...)
if ("V" %in% names(extra.args)) {
V <- extra.args$V
extra.args$V <- NULL
if (isTRUE(all.equal(dim(V), c(length(dat), 1)))) {
Vprovided <- TRUE
}
}
if (!Vprovided) {
V <- switch(param, scale = gpd.Vfun(par = mle, dat = dat), shape = gpd.Vfun(par = mle,
dat = dat), quant = gpdr.Vfun(par = mle, dat = dat, m = 1/p), VaR = gpdr.Vfun(par = mle,
dat = dat, m = m), ES = gpde.Vfun(par = mle, dat = dat, m = m), Nmean = gpdN.Vfun(par = mle,
dat = dat, N = N), Nquant = gpdr.Vfun(par = mle, dat = dat, m = 1/(1 - q^(1/N))))
}
# Obtained constrained maximum likelihood estimates for given value of psi
if (param == "scale") {
maxll <- gpd.ll(mle, dat = dat)
std.error <- sqrt(solve(gpd.infomat(par = mle, dat = dat, method = "exp"))[1, 1])
constr.mle.scale <- function(sigmat) {
as.vector(suppressWarnings(optim(par = 0.01,
fn = function(par, scale){
-gpd.ll(par = c(scale, par), dat = dat)},
method = "Brent", lower = max(-1, -sigmat/xmax),
upper = min(10, sigmat/xmin), scale = sigmat)$par))
}
# Missing psi vector
if (missing(psi) || any(is.null(psi)) || any(is.na(psi))) {
psirangelow <- pmax(1e-05, seq(-3, -1.5, length = 6) * std.error + mle[1])
lowvals <- sapply(psirangelow, function(par) {
gpd.ll(c(par, constr.mle.scale(par)), dat = dat)
}) - maxll
psirangehigh <- seq(2.5, 4, length = 6) * std.error + mle[1]
highvals <- sapply(psirangehigh, function(par) {
gpd.ll(c(par, constr.mle.scale(par)), dat = dat)
}) - maxll
lo <- approx(x = lowvals, y = psirangelow, xout = -4)$y
hi <- approx(x = highvals, y = psirangehigh, xout = -4)$y
lo <- ifelse(is.na(lo), lm(psirangelow ~ lowvals)$coef[2] * -4 + mle[1], lo)
hi <- ifelse(is.na(hi), lm(psirangehigh ~ highvals)$coef[2] * -4 + mle[1], hi)
psi <- seq(lo, hi, length = 55)
}
if (any(as.vector(psi) < 0)) {
warning("Negative scale values provided.")
psi <- psi[psi > 0]
if (length(psi) == 0) {
psi <- mle[1]
}
}
pars <- cbind(psi, sapply(psi, constr.mle.scale))
# Profile log likelihood values for psi
profll <- apply(pars, 1, function(par) {
gpd.ll(par = par, dat = dat)
})
r <- sign(mle[param] - psi) * sqrt(2 * (maxll - profll))
if ("tem" %in% mod) {
phi.mle <- gpd.phi(par = mle, dat = dat, V = V)
q2num <- apply(pars, 1, function(par) {
det(rbind(c(c(phi.mle) - gpd.phi(par = par, dat = dat, V = V)), gpd.dphi(par = par,
dat = dat, V = V)[-1, ]))
})
if (isTRUE(any(sign(q2num) * sign(r) < 0, na.rm = TRUE))) {
warning("Correction factor and likelihood root are of opposite sign - check output")
}
logq <- apply(pars, 1, function(par) {
-0.5 * log(gpd.infomat(par = par, dat = dat, method = "obs")[-1, -1])
}) + log(abs(q2num))
qmlecontrib <- -log(det(gpd.dphi(par = mle, dat = dat, V = V))) + 0.5 * log(det(gpd.infomat(par = mle,
dat = dat, method = "obs")))
logq <- logq + qmlecontrib
qcor <- sign(q2num) * exp(logq)
rstar <- ifelse(r == 0, 0, r + (logq - log(abs(r)))/r)
tem.max.opt <- function(psi, dat = dat) {
para <- c(psi, constr.mle.scale(psi))
pll <- gpd.ll(par = para, dat = dat)
rs <- 2 * (maxll - pll)
logq <- -0.5 * log(gpd.infomat(par = para, dat = dat, method = "obs")[-1, -1]) +
qmlecontrib + log(abs(det(rbind(c(c(phi.mle) - gpd.phi(par = para, dat = dat,
V = V)), gpd.dphi(par = para, dat = dat, V = V)[-1, ]))))
rs + logq - log(sqrt(abs(rs)))
}
tem.max <- optim(par = mle[1], fn = tem.max.opt, method = "Brent", dat = dat, lower = max(1e-05,
mle[1] - std.error), upper = mle[1] + std.error, control = list(abstol = 1e-10))$par
}
if ("modif" %in% mod) {
# Tangent exponential model approximation of Fraser and Reid to the profile likelihood
tem.objfunc.scale <- function(par) {
0.5 * log(gpd.infomat(par = par, dat = dat, method = "obs")[2, 2]) -
log(abs(gpd.dphi(par = par, dat = dat, V = V[, 2, drop = FALSE])[2, 1]))
}
optim.tem.fn.scale <- function(psi) {
theta.psi.opt <- constr.mle.scale(psi)
param <- c(psi, theta.psi.opt) #ll <- -theta.psi.opt$nllh
ll <- gpd.ll(param, dat = dat)
ll + tem.objfunc.scale(param)
}
# TEM profile log likelihood values for psi
proflltem <- profll + apply(pars, 1, tem.objfunc.scale)
# Maximum objective function for TEM (line search in neighborhood of the MLE)
tem.mle.opt <- optim(par = mle[1], fn = optim.tem.fn.scale, method = "Brent", lower = max(1e-05,
mle[1] - std.error), upper = mle[1] + std.error, control = list(fnscale = -1))
tem.mle <- c(tem.mle.opt$par, constr.mle.scale(tem.mle.opt$par))
# Severini empirical covariance function adjustment to profile likelihood Score function -
# observation-wise
gpd.score.f <- function(par, dat) {
sigma = par[1]
xi = par[2]
if (!isTRUE(all.equal(0, xi))) {
cbind(dat * (xi + 1)/(sigma^2 * (dat * xi/sigma + 1)) - 1/sigma, -dat * (1/xi +
1)/(sigma * (dat * xi/sigma + 1)) + log(pmax(dat * xi/sigma + 1, 0))/xi^2)
} else {
cbind((dat - sigma)/sigma^2, 1/2 * (dat - 2 * sigma) * dat/sigma^2)
}
}
# Score at MLE (sums to zero)
score.scale.mle <- gpd.score.f(mle, dat)[, 2] #keep s_lambda
empcov.objfunc.scale <- function(par) {
0.5 * log(gpd.infomat(par = par, dat = dat, method = "obs")[2, 2]) -
log(abs(sum(score.scale.mle * gpd.score.f(par, dat)[, 2])))
}
profllempcov <- profll + apply(pars, 1, empcov.objfunc.scale)
optim.empcov.fn.scale <- function(psi) {
theta.psi.opt <- constr.mle.scale(psi)
param <- c(psi, theta.psi.opt)
ll <- gpd.ll(param, dat = dat)
ll + empcov.objfunc.scale(param)
}
empcov.mle.opt <- optim(par = mle[1], fn = optim.empcov.fn.scale, method = "Brent",
lower = max(1e-05, mle[1] - std.error), upper = mle[1] + std.error, control = list(fnscale = -1))
empcov.mle <- c(empcov.mle.opt$par, constr.mle.scale(empcov.mle.opt$par))
}
# Shape parameter
} else if (param == "shape") {
maxll <- gpd.ll(mle, dat = dat)
std.error <- sqrt(solve(gpd.infomat(par = mle, dat = dat, method = "exp"))[2, 2])
constr.mle.shape <- function(xit) {
as.vector(suppressWarnings(optim(par = 2 * abs(xit) * xmax, fn = function(par, shape){-gpd.ll(par = c(par, shape), dat = dat)},
method = "Brent", shape = xit, lower = ifelse(xit < 0, abs(xit) * xmax + 1e-05, 1e-05), upper = 1e+10)$par))
}
# Missing psi vector
if (missing(psi) || any(is.null(psi)) || any(is.na(psi))) {
psirangelow <- seq(ifelse(mle[2] < 0, -7, -5), -1.5, length = 10) * std.error + mle[2]
psirangelow <- psirangelow[psirangelow > -1]
lowvals <- sapply(psirangelow, function(par) {
gpd.ll(c(constr.mle.shape(par), par), dat = dat)
}) - maxll
psirangehigh <- seq(1.5, 10, length = 10) * std.error + mle[2]
highvals <- sapply(psirangehigh, function(par) {
gpd.ll(c(constr.mle.shape(par), par), dat = dat)
}) - maxll
lo <- approx(x = lowvals, y = psirangelow, xout = -4)$y
hi <- approx(x = highvals, y = psirangehigh, xout = -4)$y
lo <- ifelse(is.na(lo), lm(psirangelow ~ lowvals)$coef[2] * -4 + mle[2], lo)
hi <- ifelse(is.na(hi), lm(psirangehigh ~ highvals)$coef[2] * -4 + mle[2], hi)
psi <- seq(lo, hi, length = 55)
psi <- psi[psi>-1]
}
pars <- cbind(sapply(psi, constr.mle.shape), psi)
# Profile log likelihood values for psi
profll <- apply(pars, 1, function(par) {
gpd.ll(par = par, dat = dat)
})
r <- sign(mle[param] - psi) * sqrt(2 * (maxll - profll))
if ("tem" %in% mod) {
phi.mle <- gpd.phi(par = mle, dat = dat, V = V)
q2num <- apply(pars, 1, function(par) {
det(rbind(gpd.dphi(par = par, dat = dat, V = V)[-2, ], c(c(phi.mle) - gpd.phi(par = par,
dat = dat, V = V))))
})
if (isTRUE(any(sign(q2num) * sign(r) < 0, na.rm = TRUE))) {
warning("Correction factor and likelihood root are of opposite sign - check output")
}
logq <- apply(pars, 1, function(par) {
-0.5 * log(gpd.infomat(par = par, dat = dat, method = "obs")[-2, -2])
}) + log(abs(q2num))
qmlecontrib <- -log(det(gpd.dphi(par = mle, dat = dat, V = V))) + 0.5 * log(det(gpd.infomat(par = mle,
dat = dat, method = "obs")))
logq <- logq + qmlecontrib
qcor <- sign(q2num) * exp(logq)
rstar <- ifelse(r == 0, 0, r + (logq - log(abs(r)))/r)
# Maximum of TEM likelihood - indirect estimation via rstar
tem.max.opt <- function(psi, dat = dat) {
para <- c(constr.mle.shape(psi), psi)
pll <- gpd.ll(par = para, dat = dat)
rs <- 2 * (maxll - pll)
logq <- -0.5 * log(gpd.infomat(par = para, dat = dat, method = "obs")[-2, -2]) +
qmlecontrib + log(abs(det(rbind(c(c(phi.mle) - gpd.phi(par = para, dat = dat,
V = V)), gpd.dphi(par = para, dat = dat, V = V)[-2, ]))))
rs + logq - log(sqrt(abs(rs)))
}
tem.max <- optim(par = mle[2], fn = tem.max.opt, method = "Brent", dat = dat, lower = mle[2] -
std.error, upper = mle[2] + std.error, control = list(abstol = 1e-10))$par
}
if ("modif" %in% mod) {
# Tangent exponential model approximation of Fraser and Reid to the profile likelihood
tem.objfunc.shape <- function(par) {
0.5 * log(gpd.infomat(par = par, dat = dat, method = "obs")[1, 1]) -
log(abs(gpd.dphi(par = par, dat = dat, V = V[, 1, drop = FALSE])[1, 1]))
}
optim.tem.fn.shape <- function(psi) {
theta.psi.opt <- constr.mle.shape(psi)
param <- c(theta.psi.opt, psi)
ll <- gpd.ll(param, dat = dat)
ll + tem.objfunc.shape(param)
}
# TEM profile log likelihood values for psi
proflltem <- profll + apply(pars, 1, tem.objfunc.shape)
# Maximum objective function for TEM (line search in neighborhood of the MLE)
tem.mle.opt <- optim(par = mle[2], fn = optim.tem.fn.shape, method = "Brent", lower = mle[2] -
std.error, upper = mle[2] + std.error, control = list(fnscale = -1))
tem.mle <- c(constr.mle.shape(tem.mle.opt$par), tem.mle.opt$par)
# Severini empirical covariance function adjustment to profile likelihood Score function -
# observation-wise
gpd.score.f <- function(par, dat) {
sigma = par[1]
xi = par[2]
if (!isTRUE(all.equal(0, xi))) {
cbind(dat * (xi + 1)/(sigma^2 * (dat * xi/sigma + 1)) - 1/sigma, -dat * (1/xi +
1)/(sigma * (dat * xi/sigma + 1)) + log(pmax(dat * xi/sigma + 1, 0))/xi^2)
} else {
cbind((dat - sigma)/sigma^2, 1/2 * (dat - 2 * sigma) * dat/sigma^2)
}
}
# Score at MLE (sums to zero)
score.shape.mle <- gpd.score.f(mle, dat)[, 1] #keep s_lambda
empcov.objfunc.shape <- function(par) {
0.5 * log(gpd.infomat(par = par, dat = dat, method = "obs")[1, 1]) -
log(abs(sum(score.shape.mle * gpd.score.f(par, dat)[, 1])))
}
profllempcov <- profll + apply(pars, 1, empcov.objfunc.shape)
optim.empcov.fn.shape <- function(psi) {
theta.psi.opt <- constr.mle.shape(psi)
param <- c(theta.psi.opt, psi)
ll <- gpd.ll(param, dat = dat)
ll + empcov.objfunc.shape(param)
}
empcov.mle.opt <- optim(par = mle[2], fn = optim.empcov.fn.shape, method = "Brent",
lower = mle[2] - std.error, upper = mle[2] + std.error, control = list(fnscale = -1))
empcov.mle <- c(constr.mle.shape(empcov.mle.opt$par), empcov.mle.opt$par)
}
# Return levels, quantiles or value-at-risk
} else if (param %in% c("quant", "VaR", "Nquant")) {
if (param == "quant") {
m <- 1/p
}
if (param == "Nquant") {
m <- 1/(1 - q^(1/N))
}
maxll <- gpdr.ll(mle, dat = dat, m = m)
std.error <- sqrt(solve(gpdr.infomat(par = mle, dat = dat, method = "exp", m = m))[1,
1])
constr.mle.quant <- function(quant) {
suppressMessages(Rsolnp::solnp(par = 0.01, function(lambda, psi, m) {
-gpdr.ll(par = c(psi, lambda), dat = dat, m = m)
}, psi = quant, m = m, lower = -1, upper = 5)$par)
}
# Missing psi vector
if (missing(psi) || any(is.null(psi)) || any(is.na(psi))) {
psirangelow <- unique(pmax(mean(dat), seq(-3, -0.5, length = 12) * std.error +
mle[1]))
lowvals <- sapply(psirangelow, function(par) {
gpdr.ll(c(par, constr.mle.quant(par)), m = m, dat = dat)
}) - maxll
psirangehigh <- seq(2, 10, length = 12) * std.error + mle[1]
highvals <- sapply(psirangehigh, function(par) {
gpdr.ll(c(par, constr.mle.quant(par)), m = m, dat = dat)
}) - maxll
lo <- approx(x = lowvals, y = psirangelow, xout = -4)$y
hi <- approx(x = highvals, y = psirangehigh, xout = -4)$y
lo <- ifelse(is.na(lo), lm(psirangelow ~ lowvals)$coef[2] * -4 + mle[1], lo)
hi <- ifelse(is.na(hi), lm(psirangehigh ~ highvals)$coef[2] * -4 + mle[1], hi)
psi <- seq(lo, hi, length = 55)
} else{
psi <- psi - threshold
if(any(psi < 0)){
stop("Invalid psi sequence: rescaled psi for quantiles must be positive")
}
}
pars <- cbind(psi, sapply(psi, constr.mle.quant))
# Profile log likelihood values for psi
profll <- apply(pars, 1, function(par) {
gpdr.ll(par = par, dat = dat, m = m)
})
r <- sign(mle[param] - psi) * sqrt(2 * (maxll - profll))
if ("tem" %in% mod) {
phi.mle <- gpdr.phi(par = mle, dat = dat, m = m, V = V)
q2num <- apply(pars, 1, function(par) {
det(rbind(c(c(phi.mle) - gpdr.phi(par = par, dat = dat, V = V, m = m)), gpdr.dphi(par = par,
dat = dat, V = V, m = m)[-1, ]))
})
if (isTRUE(any(sign(q2num) * sign(r) < 0, na.rm = TRUE))) {
warning("Correction factor and likelihood root are of opposite sign - check output")
}
logq <- apply(pars, 1, function(par) {
-0.5 * log(gpdr.infomat(par = par, dat = dat, method = "obs", m = m)[-1, -1])
}) + log(abs(q2num))
qmlecontrib <- -log(det(gpdr.dphi(par = mle, dat = dat, V = V, m = m))) + 0.5 *
log(det(gpdr.infomat(par = mle, dat = dat, method = "obs", m = m)))
logq <- logq + qmlecontrib
qcor <- sign(q2num) * exp(logq)
rstar <- ifelse(r == 0, 0, r + (logq - log(abs(r)))/r)
tem.max.opt <- function(psi, dat = dat) {
para <- c(psi, constr.mle.quant(psi))
pll <- gpdr.ll(par = para, dat = dat, m = m)
rs <- 2 * (maxll - pll)
logq <- -0.5 * log(gpdr.infomat(par = para, dat = dat, method = "obs", m = m)[-1,
-1]) + qmlecontrib + log(abs(det(rbind(c(c(phi.mle) - gpdr.phi(par = para,
dat = dat, V = V, m = m)), gpdr.dphi(par = para, dat = dat, V = V, m = m)[-1,
]))))
rs + logq - log(sqrt(abs(rs)))
}
tem.max <- optim(par = mle[1], fn = tem.max.opt, method = "Brent", dat = dat, lower = max(1e-05,
mle[1] - std.error), upper = mle[1] + std.error, control = list(abstol = 1e-10))$par
}
if ("modif" %in% mod) {
# Tangent exponential model approximation of Fraser and Reid to the profile likelihood
tem.objfunc.quant <- function(par) {
0.5 * log(gpdr.infomat(par = par, dat = dat, method = "obs", m = m)[2, 2]) -
log(abs(gpdr.dphi(par = par, dat = dat, m = m, V = V[, 2, drop = FALSE])[2, 1]))
}
optim.tem.fn.quant <- function(psi) {
theta.psi.opt <- constr.mle.quant(psi)
param <- c(psi, theta.psi.opt)
ll <- gpdr.ll(param, dat = dat, m = m)
ll + tem.objfunc.quant(param)
}
# TEM profile log likelihood values for psi
proflltem <- profll + suppressWarnings(apply(pars, 1, tem.objfunc.quant))
# Maximum objective function for TEM
tem.mle.opt <- optim(par = mle[1], fn = optim.tem.fn.quant, method = "Brent", lower = max(1e-05,
mle[1] - std.error), upper = mle[1] + std.error, control = list(fnscale = -1))
tem.mle <- c(tem.mle.opt$par, constr.mle.quant(tem.mle.opt$par))
# Severini empirical covariance function adjustment to profile likelihood Score function -
# observation-wise for xi
gpdr.score.f <- function(par, dat, m) {
xi = par[2]
r = par[1]
-dat * m^xi * (1/xi + 1) * log(m)/((dat * (m^xi - 1)/r + 1) * r) + (m^xi *
r * xi * log(m)/(m^xi - 1)^2 - r/(m^xi - 1)) * (m^xi - 1)/(r * xi) + log(dat *
(m^xi - 1)/r + 1)/xi^2
}
# Score at MLE (sums to zero)
score.quant.mle <- gpdr.score.f(mle, dat, m)
empcov.objfunc.quant <- function(par) {
0.5 * log(gpdr.infomat(par = par, dat = dat, method = "obs", m = m)[2, 2]) -
log(abs(sum(score.quant.mle * gpdr.score.f(par, dat, m = m))))
}
profllempcov <- profll + suppressWarnings(apply(pars, 1, empcov.objfunc.quant))
optim.empcov.fn.quant <- function(psi) {
theta.psi.opt <- constr.mle.quant(psi)
param <- c(psi, theta.psi.opt)
ll <- gpdr.ll(param, dat = dat, m = m)
ll + empcov.objfunc.quant(param)
}
empcov.mle.opt <- optim(par = mle[1], fn = optim.empcov.fn.quant, method = "Brent",
lower = max(1e-05, mle[1] - std.error), upper = mle[1] + std.error, control = list(fnscale = -1))
empcov.mle <- c(empcov.mle.opt$par, constr.mle.quant(empcov.mle.opt$par))
}
} else if (param == "ES") {
maxll <- gpde.ll(mle, dat = dat, m = m)
std.error <- sqrt(solve(gpde.infomat(par = mle, dat = dat, method = "exp", m = m))[1,
1])
constr.mle.es <- function(psif) {
# nloptr::auglag(x0 = 0.1, fn = function(x){-gpde.ll(par = c(psif, x), dat = dat, m = m)},
# hin = function(x){ c(psif*(1-x)*((m^x-1)/x+1)^(-1),
# 1+psif*(1-x)*((m^x-1)/x+1)^(-1)/x*xmin, 1+psif*(1-x)*((m^x-1)/x+1)^(-1)/x*xmax)}, lower =
# -1+1e-10, upper = 1-1e-5)$par}
optim(par = 0.1, fn = function(x) {
-gpde.ll(par = c(psif, x), dat = dat, m = m)
}, method = "Brent", lower = -1 + 1e-10, upper = 1 - 1e-05)$par
}
# Missing psi vector
if (missing(psi) || any(is.null(psi)) || any(is.na(psi))) {
psirangelow <- unique(pmax(mean(dat), seq(-3, -0.1, length = 15) * std.error +
mle[1]))
lowvals <- sapply(psirangelow, function(par) {
gpde.ll(c(par, constr.mle.es(par)), m = m, dat = dat)
}) - maxll
psirangehigh <- seq(2.5, 30, length = 12) * std.error + mle[1]
highvals <- sapply(psirangehigh, function(par) {
gpde.ll(c(par, constr.mle.es(par)), m = m, dat = dat)
}) - maxll
lo <- approx(x = lowvals, y = psirangelow, xout = -4)$y
hi <- approx(x = highvals, y = psirangehigh, xout = -4)$y
lo <- ifelse(is.na(lo), lm(psirangelow ~ lowvals)$coef[2] * -4 + mle[1], lo)
hi <- ifelse(is.na(hi), lm(psirangehigh ~ highvals)$coef[2] * -4.5 + mle[1], hi)
psi <- c(seq(lo, mle[1], length = 15)[-15], seq(mle[1], min(mle[1] + 2 * std.error,
hi), length = 25)[-1])
if (mle[1] + 2 * std.error < hi) {
psi <- c(psi, seq(mle[1] + 2 * std.error, hi, length = 20))
}
}
# Do not remove threshold for expected shortfall
pars <- cbind(psi, sapply(psi, constr.mle.es))
# Profile log likelihood values for psi
profll <- apply(pars, 1, function(par) {
gpde.ll(par = par, dat = dat, m = m)
})
profll[profll == 1e+10] <- NA
r <- sign(mle[param] - psi) * sqrt(2 * (maxll - profll))
if ("tem" %in% mod) {
phi.mle <- gpde.phi(par = mle, dat = dat, m = m, V = V)
q2num <- apply(pars, 1, function(par) {
det(rbind(c(c(phi.mle) - gpde.phi(par = par, dat = dat, V = V, m = m)), gpde.dphi(par = par,
dat = dat, V = V, m = m)[-1, ]))
})
if (isTRUE(any(sign(q2num) * sign(r) < 0, na.rm = TRUE))) {
warning("Correction factor and likelihood root are of opposite sign - check output")
}
logq <- apply(pars, 1, function(par) {
-0.5 * log(gpde.infomat(par = par, dat = dat, method = "obs", m = m)[-1, -1])
}) + log(abs(q2num))
qmlecontrib <- -log(det(gpde.dphi(par = mle, dat = dat, V = V, m = m))) + 0.5 *
log(det(gpde.infomat(par = mle, dat = dat, method = "obs", m = m)))
logq <- logq + qmlecontrib
qcor <- sign(q2num) * exp(logq)
rstar <- ifelse(r == 0, 0, r + (logq - log(abs(r)))/r)
tem.max.opt <- function(psi, dat = dat) {
para <- c(psi, constr.mle.es(psi))
pll <- gpde.ll(par = para, dat = dat, m = m)
rs <- 2 * (maxll - pll)
logq <- -0.5 * log(gpde.infomat(par = para, dat = dat, method = "obs", m = m)[-1,
-1]) + qmlecontrib + log(abs(det(rbind(c(c(phi.mle) - gpde.phi(par = para,
dat = dat, V = V, m = m)), gpde.dphi(par = para, dat = dat, V = V, m = m)[-1,
]))))
rs + logq - log(sqrt(abs(rs)))
}
tem.max <- optim(par = mle[1], fn = tem.max.opt, method = "Brent", dat = dat, lower = max(1e-05,
mle[1] - std.error), upper = mle[1] + std.error, control = list(abstol = 1e-10))$par
}
if ("modif" %in% mod) {
# Tangent exponential model approximation of Fraser and Reid to the profile likelihood
tem.objfunc.es <- function(par) {
0.5 * log(gpde.infomat(par = par, dat = dat, method = "obs", m = m)[2, 2]) -
log(abs(gpde.dphi(par = par, dat = dat, m = m, V = V[, 2, drop = FALSE])[2, 1]))
}
optim.tem.fn.es <- function(psi) {
theta.psi.opt <- constr.mle.es(psi)
param <- c(psi, theta.psi.opt)
ll <- gpde.ll(param, dat = dat, m = m)
ll + tem.objfunc.es(param)
}
# TEM profile log likelihood values for psi
proflltem <- profll + suppressWarnings(apply(pars, 1, tem.objfunc.es))
# Maximum objective function for TEM
tem.mle.opt <- optim(par = mle[1], fn = optim.tem.fn.es, method = "Brent", lower = max(quantile(dat,
1 - 1/m), mle[1] - std.error), upper = mle[1] + std.error, control = list(fnscale = -1))
tem.mle <- c(tem.mle.opt$par, constr.mle.es(tem.mle.opt$par))
# Severini empirical covariance function adjustment to profile likelihood Score function -
# observation-wise for xi
gpde.score.f <- function(par, dat, m) {
xi = par[2]
es = par[1]
-((m^xi * log(m) + 1) * dat/(es * (xi - 1)) - dat * (m^xi + xi - 1)/(es * (xi -
1)^2)) * (1/xi + 1)/(dat * (m^xi + xi - 1)/(es * (xi - 1)) - 1) + (m^xi +
xi - 1) * ((m^xi * log(m) + 1) * (xi - 1) * xi/(m^xi + xi - 1)^2 - (xi -
1)/(m^xi + xi - 1) - xi/(m^xi + xi - 1))/((xi - 1) * xi) + log(pmax(0, -dat *
(m^xi + xi - 1)/(es * (xi - 1)) + 1))/xi^2
}
# Score at MLE (sums to zero)
score.es.mle <- gpde.score.f(mle, dat, m)
empcov.objfunc.es <- function(par) {
0.5 * log(gpde.infomat(par = par, dat = dat, method = "obs", m = m)[2, 2]) -
log(abs(sum(score.es.mle * gpde.score.f(par, dat, m = m))))
}
profllempcov <- profll + suppressWarnings(apply(pars, 1, empcov.objfunc.es))
optim.empcov.fn.es <- function(psi) {
theta.psi.opt <- constr.mle.es(psi)
param <- c(psi, theta.psi.opt)
ll <- gpde.ll(param, dat = dat, m = m)
ll + empcov.objfunc.es(param)
}
empcov.mle.opt <- optim(par = mle[1], fn = optim.empcov.fn.es, method = "Brent",
lower = max(quantile(dat, 1 - 1/m), mle[1] - std.error), upper = mle[1] + std.error,
control = list(fnscale = -1))
empcov.mle <- c(empcov.mle.opt$par, constr.mle.es(empcov.mle.opt$par))
}
} else if (param == "Nmean") {
maxll <- gpdN.ll(mle, dat = dat, N = N)
std.error <- sqrt(solve(gpdN.infomat(par = mle, dat = dat, method = "exp", N = N))[1])
constr.mle.Nmean <- function(Nmeant) {
suppressWarnings(alabama::auglag(par = 0.01, function(lambda, psi, N) {
-gpdN.ll(par = c(psi, lambda), dat = dat, N = N)
}, gr = function(lambda, psi, N) {
-gpdN.score(par = c(psi, lambda), dat = dat, N = N)[2]
},
hin = function(lambda, psi, N){
sigma = ifelse(abs(lambda > 1e-8),
psi * lambda/(exp(lgamma(N + 1) + lgamma(-lambda + 1) - lgamma(N - lambda + 1)) - 1),
psi / (0.57721566490153231044 + psigamma(N + 1)))
if(lambda >= 0){c(1e-8, sigma, 1-lambda, lambda + 1)} else{ c(- sigma/lambda - xmax, sigma, 1 - lambda, lambda + 1)}},
psi = Nmeant, N = N, control.outer = list(trace = FALSE))$par)
}
# Missing psi vector
if (missing(psi) || any(is.null(psi)) || any(is.na(psi))) {
#compute profile log-lik on a grid left and right of the MLE
psirangelow <- unique(pmax(mean(dat), seq(-3, -0.25, length = 20) * std.error +
mle[1]))
lowvals <- sapply(psirangelow, function(par) {
gpdN.ll(c(par, constr.mle.Nmean(par)), dat = dat, N = N)
}) - maxll
psirangehigh <- seq(2.5, 50, length = 20) * std.error + mle[1]
highvals <- sapply(psirangehigh, function(par) {
gpdN.ll(c(par, constr.mle.Nmean(par)), dat = dat, N = N)
}) - maxll
#Try to do linear interpolation - only works if value inside the interval lowvals or highvals
lo <- approx(x = lowvals, y = psirangelow, xout = -4)$y
#Else linear interpolation with linear model fitted to lower values
lo <- ifelse(is.na(lo), spline(x = lowvals, y = psirangelow, xout = -4)$y, lo)
psirangelow <- seq(lo, mle[1], length = 20)
lowvals <- sapply(psirangelow, function(par) {
gpdN.ll(c(par, constr.mle.Nmean(par)), dat = dat, N = N)
}) - maxll
#hi <- approx(x = highvals, y = psirangehigh, xout = -4)$y
#For upper, use spline approx
hi <- spline(x = highvals, y = psirangehigh, xout = -4)$y
#Recompute the range
psirangehigh <- seq(psirangehigh[1], hi, length = 30)
highvals <- sapply(psirangehigh, function(par) {
gpdN.ll(c(par, constr.mle.Nmean(par)), dat = dat, N = N)
}) - maxll
#Remove NAs, inf, etc.
highvals <- highvals[is.finite(highvals)]
psirangehigh <- psirangehigh[1:length(highvals)]
#If could not interpolate, use a simple linear model to predict lower value
#hi <- ifelse(is.na(hi), spline(x = highvals, y = psirangehigh, xout = -4)$y, hi)
psi <- as.vector(c(spline(x=c(lowvals, 0), y = c(psirangelow, mle[1]), xout = seq(-4, -0.1, length = 15))$y, mle[1],
spline(x=c(0, highvals), y = c(mle[1], psirangehigh), xout = seq(-0.1, highvals[length(highvals)], length = 20))$y))
} else{
psi <- psi - threshold
}
if (any(as.vector(psi) < 0)) {
warning("Negative Nmean values provided.")
psi <- psi[psi > 0]
if (length(psi) == 0) {
psi <- mle[1]
}
}
pars <- cbind(psi, sapply(psi, constr.mle.Nmean))
# Profile log likelihood values for psi
profll <- apply(pars, 1, function(par) {
gpdN.ll(par = par, dat = dat, N = N)
})
r <- sign(mle[param] - psi) * sqrt(2 * (maxll - profll))
if ("tem" %in% mod) {
phi.mle <- gpdN.phi(par = mle, dat = dat, N = N, V = V)
q2num <- apply(pars, 1, function(par) {
det(rbind(c(c(phi.mle) - gpdN.phi(par = par, dat = dat, V = V, N = N)), gpdN.dphi(par = par,
dat = dat, V = V, N = N)[-1, ]))
})
if (isTRUE(any(sign(q2num) * sign(r) < 0, na.rm = TRUE))) {
warning("Correction factor and likelihood root are of opposite sign - check output")
}
logq <- apply(pars, 1, function(par) {
-0.5 * log(gpdN.infomat(par = par, dat = dat, method = "obs", N = N)[-1, -1])
}) + log(abs(q2num))
qmlecontrib <- -log(det(gpdN.dphi(par = mle, dat = dat, V = V, N = N))) + 0.5 *
log(det(gpdN.infomat(par = mle, dat = dat, method = "obs", N = N)))
logq <- logq + qmlecontrib
qcor <- sign(q2num) * exp(logq)
rstar <- ifelse(r == 0, 0, r + (logq - log(abs(r)))/r)
tem.max.opt <- function(psi, dat = dat) {
para <- c(psi, constr.mle.Nmean(psi))
pll <- gpdN.ll(par = para, dat = dat, N = N)
rs <- 2 * (maxll - pll)
logq <- -0.5 * log(gpdN.infomat(par = para, dat = dat, method = "obs", N = N)[-1,
-1]) + qmlecontrib + log(abs(det(rbind(c(c(phi.mle) - gpdN.phi(par = para,
dat = dat, V = V, N = N)), gpdN.dphi(par = para, dat = dat, V = V, N = N)[-1,
]))))
rs + logq - log(sqrt(abs(rs)))
}
tem.max <- optim(par = mle[1], fn = tem.max.opt, method = "Brent", dat = dat, lower = max(1e-05,
mle[1] - std.error), upper = mle[1] + std.error, control = list(abstol = 1e-10))$par
}
if ("modif" %in% mod) {
# Tangent exponential model approximation of Fraser and Reid to the profile likelihood
tem.objfunc.Nmean <- function(par) {
0.5 * log(gpdN.infomat(par = par, dat = dat, method = "obs", N = N)[2, 2]) -
log(abs(gpdN.dphi(par = par, dat = dat, N = N, V = V[, 2, drop = FALSE])[2, 1]))
}
optim.tem.fn.Nmean <- function(psi) {
theta.psi.opt <- constr.mle.Nmean(psi)
param <- c(psi, theta.psi.opt)
ll <- gpdN.ll(param, dat = dat, N = N)
ll + tem.objfunc.Nmean(param)
}
# TEM profile log likelihood values for psi
proflltem <- profll + suppressWarnings(apply(pars, 1, tem.objfunc.Nmean))
# Maximum objective function for TEM
tem.mle.opt <- optim(par = mle[1], fn = optim.tem.fn.Nmean, method = "Brent", lower = max(1e-05,
mle[1] - std.error), upper = mle[1] + std.error, control = list(fnscale = -1))
tem.mle <- c(tem.mle.opt$par, constr.mle.Nmean(tem.mle.opt$par))
# Severini empirical covariance function adjustment to profile likelihood Score function -
# observation-wise for xi
gpdN.score.f <- function(par, dat, N) {
z = par[1]
xi = par[2]
cst <- exp(lgamma(N + 1) + lgamma(1 - xi) - lgamma(N + 1 - xi))
-(psigamma(N - xi + 1) * cst - psigamma(-xi + 1) * cst) * dat * (1/xi + 1)/(z *
(dat * (cst - 1)/z + 1)) + ((psigamma(N - xi + 1) * cst - psigamma(-xi +
1) * cst) * xi * z/(cst - 1)^2 - z/(cst - 1)) * (cst - 1)/(xi * z) + log(dat *
(cst - 1)/z + 1)/xi^2
}
# Score at MLE (sums to zero)
score.Nmean.mle <- gpdN.score.f(mle, dat, N)
empcov.objfunc.Nmean <- function(par) {
0.5 * log(gpdN.infomat(par = par, dat = dat, method = "obs", N = N)[2, 2]) -
log(abs(sum(score.Nmean.mle * gpdN.score.f(par, dat, N = N))))
}
profllempcov <- profll + suppressWarnings(apply(pars, 1, empcov.objfunc.Nmean))
optim.empcov.fn.Nmean <- function(psi) {
theta.psi.opt <- constr.mle.Nmean(psi)
param <- c(psi, theta.psi.opt)
ll <- gpdN.ll(param, dat = dat, N = N)
ll + empcov.objfunc.Nmean(param)
}
empcov.mle.opt <- optim(par = mle[1], fn = optim.empcov.fn.Nmean, method = "Brent",
lower = max(1e-05, mle[1] - std.error), upper = mle[1] + std.error, control = list(fnscale = -1))
empcov.mle <- c(empcov.mle.opt$par, constr.mle.Nmean(empcov.mle.opt$par))
}
}
# Return profile likelihood and quantities of interest (modified likelihoods)
colnames(pars) <- names(mle)
ans <- list(mle = mle, pars = pars, psi.max = as.vector(mle[param]), param = param, std.error = std.error,
psi = psi, pll = profll, maxpll = maxll, r = r)
# Shift by threshold if non-null
if(shiftres){
ans$psi <- ans$psi + threshold
ans$mle[1] <- ans$psi.max <- ans$mle[1] + threshold
ans$pars[,1] <- ans$pars[,1] + threshold
}
if ("tem" %in% mod) {
ans$q <- qcor
ans$rstar <- rstar
ans$tem.psimax <- as.vector(tem.max) + ifelse(shiftres, threshold, 0)
ans$normal <- c(ans$psi.max, ans$std.error)
if (correction && length(psi) > 10) {
ans <- spline.corr(ans)
}
}
if ("modif" %in% mod) {
ans$tem.mle <- ifelse(param == "shape", tem.mle[2], tem.mle[1])
if(shiftres){ ans$tem.mle[1] <- ans$tem.mle[1] + threshold }
ans$tem.pll <- proflltem
ans$tem.maxpll <- as.vector(tem.mle.opt$value)
ans$empcov.mle <- ifelse(param == "shape", empcov.mle[2], empcov.mle[1])
if(shiftres){ ans$empcov.mle[1] <- ans$empcov.mle[1] + threshold }
ans$empcov.pll <- as.vector(profllempcov)
ans$empcov.maxpll <- as.vector(empcov.mle.opt$value)
}
if ("tem" %in% mod) {
class(ans) <- c("eprof", "fr")
} else {
class(ans) <- "eprof"
}
ans$family <- "gpd"
ans$threshold <- threshold
if(plot){
plot(ans)
}
return(invisible(ans))
}
#' Plot of tangent exponential model profile likelihood
#'
#' This function is adapted from the \code{plot.fr} function from the \code{hoa} package bundle.
#' It differs from the latter mostly in the placement of legends.
#'
#' @param x an object of class \code{fr} returned by \code{\link{gpd.tem}} or \code{\link{gev.tem}}.
#' @param ... further arguments to \code{plot} currently ignored. Providing a numeric vector \code{which} allows for custom selection of the plots. A logical \code{all}. See \strong{Details}.
#' @return graphs depending on argument \code{which}
#' @details Plots produced depend on the integers provided in \code{which}. \code{1} displays the Wald pivot, the likelihood root \code{r}, the modified likelihood root \code{rstar} and the likelihood modification \code{q} as functions of the parameter \code{psi}. \code{2} gives the renormalized profile log likelihood and adjusted form, with the maximum likelihood having ordinate value of zero. \code{3} provides the significance function, a transformation of \code{1}. Lastly, \code{4} plots the correction factor as a function of the likelihood root; it is a diagnostic plot aimed for detecting failure of
#' the asymptotic approximation, often due to poor numerics in a neighborhood of \code{r=0}; the function should be smooth. The function \code{\link{spline.corr}} is designed to handle this by correcting numerically unstable estimates, replacing outliers and missing values with the fitted values from the fit.
#'
#'
#' @references Brazzale, A. R., Davison, A. C. and Reid, N. (2007). \emph{Applied Asymptotics: Case Studies in Small-Sample Statistics}. Cambridge University Press, Cambridge.
#' @export
plot.fr <- function(x, ...) {
# plot a fraser-reid object
whichPlot <- c(1:4) #default
if (length(list(...)) > 0) {
if ("which" %in% names(list(...))) {
whichPlot <- list(...)$which
whichPlot <- (1:4)[c(1:4 %in% whichPlot)]
} else if ("all" %in% names(list(...))) {
if (!is.logical(all)) {
stop("Invalid \"all\" parameter")
}
if (list(...)$all) {
whichPlot <- 1:4
} else {
whichPlot <- 1:2
}
}
}
old.pars <- par(no.readonly = TRUE)
if (sum(c(1, 2, 3, 4) %in% whichPlot) > 2) {
par(mfrow = c(2, 2), mar = c(4.8, 4.8, 1, 0.1))
} else if (sum(c(1, 2, 3, 4) %in% whichPlot) == 2) {
par(mfrow = c(1, 2))
}
fr <- x
xl <- ifelse(is.null(fr$param), expression(psi), fr$param)
if (1 %in% whichPlot) {
plot(fr$psi, fr$r, type = "l", xlab = xl, ylab = "Value of pivot", ylim = c(-4, 4),
panel.first = abline(h = qnorm(c(0.005, 0.025, 0.05, 0.5, 0.95, 0.975, 0.995)),
col = "grey", lwd = 0.7), bty = "l")
lines(fr$psi, (fr$normal[1] - fr$psi)/fr$normal[2], col = "green", lwd = 1.5)
lines(fr$psi, fr$q, col = "red", lwd = 1.5)
lines(fr$psi, fr$r, lwd = 1.5)
lines(fr$psi, fr$rstar, col = "blue")
legend(x = "topright", c("Wald pivot", "lik. root", "modif. root", expression(q(psi))),
lty = c(1, 1, 1, 1), x.intersp = 0.2, lwd = 1.5, seg.len = 0.5, col = c("green",
"black", "blue", "red"), bty = "n", cex = 0.9, xjust = 1)
}
# top right: log likelihood (and adjusted version, I think?) as a function of psi
if (2 %in% whichPlot) {
plot(fr$psi, -fr$r^2/2, type = "l", xlab = xl, ylab = "Profile log likelihood", ylim = c(-8,
0), panel.first = abline(h = -qchisq(c(0.95, 0.99), df = 1)/2, col = "grey", lwd = 0.7),
lwd = 1.5, bty = "l")
lines(fr$psi, -fr$rstar^2/2, col = "blue", lwd = 1.5)
legend(x = "bottomright", c("profile","tem"),
lty = c(1, 1), x.intersp = 0.2, lwd = 1.5, seg.len = 0.5, col = c("black", "blue"),
bty = "n")
# optional: add diagnostic panels
}
if (3 %in% whichPlot) {
# lower left: plot of Phi(pivot) as a function of psi
plot(fr$psi, pnorm(fr$r), type = "l", xlab = xl, ylab = "Significance function", ylim = c(0,
1), panel.first = abline(h = c(0.025, 0.05, 0.5, 0.95, 0.975), col = "grey", lwd = 0.7),
lwd = 1.5, bty = "l")
lines(fr$psi, pnorm(fr$q), col = "red", lwd = 1.5)
lines(fr$psi, pnorm(fr$rstar), col = "blue", lwd = 1.5)
legend(x = "topright", c("lik. root", "modif. root", expression(q(psi))), lty = c(1,
1, 1), col = c("black", "blue", "red"), bty = "n", x.intersp = 0.2, lwd = 1.5,
seg.len = 0.5, cex = 0.9)
}
# lower right: log(q/r)/r as a function of r (should be smooth)
if (4 %in% whichPlot) {
fit.r <- stats::smooth.spline(x = na.omit(cbind(fr$r, fr$rstar)), cv = FALSE)
pr <- predict(fit.r, 0)$y
plot(fr$r, fr$rstar-fr$r, type = "l", xlab = "Likelihood root r", ylab = expression(paste("Correction factor log(q/r)/r")),
panel.first = {
abline(h = 0, col = "grey", lwd = 0.7)
abline(v = 0, col = "grey", lwd = 0.7)
abline(v = pr, col = "grey", lwd = 0.7, lty = 2)
}, lwd = 1.5, bty = "l")
}
par(old.pars)
}
#' Spline correction for Fraser-Reid approximations
#'
#' The tangent exponential model can be numerically unstable for values close to \eqn{r = 0}.
#' This function corrects these incorrect values, which are interpolated using splines.
#' The function takes as input an object of class \code{fr} and returns the same object with
#' different \code{rstar} values.
#' @section Warning:
#'
#' While penalized (robust) splines often do a good job at capturing and correcting for numerical outliers and \code{NA}, it
#' may also be driven by unusual values lying on the profile log-likelihood the curve or fail to detect outliers (or falsely identifying `correct' values as outliers). The user should always validate by comparing the plots of both the uncorrected (raw) output of the object with that of \code{spline.corr}.
#' @details If available, the function uses \code{cobs} from the eponym package. The latter handles constraints and smoothness penalties, and is more robust than the equivalent \code{\link[stats]{smooth.spline}}.
#'
#' @param fr an object of class \code{fr}, normally the output of \link{gpd.tem} or \link{gev.tem}.
#' @param method string for the method, either \code{cobs} (constrained robust B-spline from eponym package) or \code{smooth.spline}
#' @return an object of class \code{fr}, containing as additional arguments \code{spline} and a modified \code{rstar} argument.
#' @keywords internal
#' @export
spline.corr <- function(fr, method = c("cobs","smooth.spline")) {
# Step 1: fit a smoothing spline to rstar If fit failed for some values (for example when
# shape forced to be < 1) Remove those values
method <- match.arg(method[1], choices = c("cobs","smooth.spline"))
if (all(is.nan(fr$q)) || all(is.nan(fr$rstar))) {
# If could not compute Fraser-Reid correction, abort
return(fr)
}
fitfailed <- which(!is.finite(fr$r))
if (length(fitfailed) > 0) {
fr$r <- fr$r[-fitfailed]
fr$rstar <- fr$rstar[-fitfailed]
fr$q <- fr$q[-fitfailed]
fr$psi <- fr$psi[-fitfailed]
}
w <- pchisq(fr$r^2, 0.5)
# If any correction for q failed and returned NA
corfailed <- which(!is.finite(fr$rstar))
# If equispaced values for psi between MLE and other, then we have r = 0
corfailed <- c(corfailed, which(fr$r == 0))
if (length(corfailed) > 0) {
resp <- (fr$rstar - fr$r)[-corfailed]
regr <- fr$r[-corfailed]
w <- w[-corfailed]
} else {
resp <- (fr$rstar - fr$r)
regr <- fr$r
}
if (method == "cobs" && requireNamespace("cobs", quietly = TRUE)) {
spline <- cobs::cobs(y = resp, x = regr, w = w, constraint = "none", lambda = 1, nknots = 20,
print.mesg = FALSE, print.warn = FALSE)$fitted
} else {
spline <- rev(stats::smooth.spline(y = resp, x = regr, w = w, spar = 0.9, cv = TRUE)$y)
}
# Compute difference between fitted values and rstar
departure <- spline - resp
# Ad-hoc fix of the values close to MLE where the numerical precision causes difficulty
# Outlier detection via chi-square test From package outliers, (c)Lukasz Komsta
scores <- function(x, prob = NA) {
(x - mean(x))^2/var(x) > qchisq(prob, 1)
}
bad <- which(scores(departure, prob = 0.95))
if (length(bad) > 0) {
# Exclude those values if they are in the end of the distribution
bad <- bad[intersect(which(bad < 0.85 * length(departure)), which(bad > 0.15 * length(departure)))]
# Remove outliers and fit again (with less smoothness)
}
if (length(bad) > 0) {
resp[bad] <- NA
w <- w[-bad]
}
if (requireNamespace("cobs", quietly = TRUE)) {
spline <- cobs::cobs(y = resp, x = regr, constraint = "none", w = w, lambda = -1, ic = "SIC",
knots.add = TRUE, repeat.delete.add = TRUE, print.mesg = FALSE, print.warn = FALSE)
fr$spline <- spline
fr$rstar <- predict(spline, fr$r, interval = "none")[, 2] + fr$r
} else {
spline <- stats::smooth.spline(x = na.omit(cbind(regr, resp)), cv = FALSE, all.knots = TRUE)
fr$spline <- spline
fr$rstar <- predict(spline, fr$r)$y + fr$r
}
return(fr)
}
#' Bridging the singularity for higher order asymptotics
#'
#' The correction factor \eqn{\log(q/r)/r} for the
#' likelihood root is unbounded in the vincinity of
#' the maximum likelihood estimator. The thesis of
#' Rongcai Li (University of Toronto, 2001)
#' explores different ways of bridging this
#' singularity, notably using asymptotic expansions.
#'
#' The poor man's method used here consists in
#' fitting a robust regression to \eqn{1/q-1/r}
#' as a function of \eqn{r} and using predictions
#' from the model to solve for \eqn{q}. This
#' approach is seemingly superior to that
#' previously used in \link{spline.corr}.
#'
#' @param fran object of class \code{fr}
#' @param print.warning logical; should warning message be printed? Default to \code{FALSE}
##' @return an object of class \code{fr}, containing as additional arguments \code{spline} and a modified \code{rstar} argument.
#' @keywords internal
#' @export
tem.corr <- function(fr, print.warning = FALSE) {
if(!requireNamespace("MASS", quietly = TRUE)){
stop("The \"MASS\" package is required for this function to work")
}
if (all(is.nan(fr$q)) || all(is.nan(fr$rstar))) {
# If could not compute Fraser-Reid correction, abort
return(fr)
}
fitfailed <- which(!is.finite(fr$r))
if (length(fitfailed) > 0) {
fr$r <- fr$r[-fitfailed]
fr$rstar <- fr$rstar[-fitfailed]
fr$q <- fr$q[-fitfailed]
fr$psi <- fr$psi[-fitfailed]
}
if(length(fr$psi) < 25L){
if(print.warning){
warning("The correction for the tangent exponential model\n is based on less than 25 observations.")
}
}
# this is approximately linear for fixed data
resp <- 1/fr$q - 1/fr$r
# fit a robust regression to discount observations that are outlying
robust_reg <- MASS::rlm(resp ~ fr$r)
# Replace q values by predictions
pred <- predict(robust_reg)
qhat <- 1/(pred + 1/fr$r)
fr$q_old <- fr$q
fr$rstar_old <- fr$rstar
fr$q <- qhat
fr$rstar <- fr$r + log(fr$q/fr$r)/fr$r
fr$tem.psimax
return(fr)
}
#' Tangent exponential model approximation for the GEV distribution
#'
#' The function \code{gev.tem} provides a tangent exponential model (TEM) approximation
#' for higher order likelihood inference for a scalar parameter for the generalized extreme value distribution.
#' Options include location scale and shape parameters as well as value-at-risk (or return levels).
#' The function attempts to find good values for \code{psi} that will
#' cover the range of options, but the fail may fit and return an error.
#'
#'
#' @param param parameter over which to profile
#' @param psi scalar or ordered vector of values for the interest parameter. If \code{NULL} (default), a grid of values centered at the MLE is selected
#' @param N size of block over which to take maxima. Required only for \code{param} \code{Nmean} and \code{Nquant}.
#' @param p tail probability for the (1-p)th quantile (return levels). Required only if \code{param = 'retlev'}
#' @param q probability level of quantile. Required only for \code{param} \code{Nquant}.
#' @param dat sample vector for the GEV distribution
#' @param n.psi number of values of \code{psi} at which the likelihood is computed, if \code{psi} is not supplied (\code{NULL}). Odd values are more prone to give rise to numerical instabilities near the MLE. If \code{psi} is a vector of length 2 and \code{n.psi} is greater than 2, these are taken to be endpoints of the sequence.
#' @param plot logical indicating whether \code{plot.fr} should be called upon exit
#' @param correction logical indicating whether \link{spline.corr} should be called.
#' @author Leo Belzile
#' @return an invisible object of class \code{fr} (see \code{tem} in package \code{hoa}) with elements
#' \itemize{
#' \item{\code{normal}: }{maximum likelihood estimate and standard error of the interest parameter \eqn{psi}}
#' \item{\code{par.hat}: }{maximum likelihood estimates}
#' \item{\code{par.hat.se}: }{standard errors of maximum likelihood estimates}
#' \item{\code{th.rest}: }{estimated maximum profile likelihood at (\eqn{psi}, \eqn{\hat{\lambda}})}
#' \item{\code{r}: }{values of likelihood root corresponding to \eqn{\psi}}
#' \item{\code{psi}: }{vector of interest parameter}
#' \item{\code{q}: }{vector of likelihood modifications}
#' \item{\code{rstar}: }{modified likelihood root vector}
#' \item{\code{rstar.old}: }{uncorrected modified likelihood root vector}
#' \item{\code{param}: }{parameter}
#' }
#' @export
#' @examples
#' \dontrun{
#' set.seed(1234)
#' dat <- rgev(n = 40, loc = 0, scale = 2, shape = -0.1)
#' gev.tem('shape', dat = dat, plot = TRUE)
#' gev.tem('quant', dat = dat, p = 0.01, plot = TRUE)
#' gev.tem('scale', psi = seq(1, 4, by = 0.1), dat = dat, plot = TRUE)
#' dat <- rgev(n = 40, loc = 0, scale = 2, shape = 0.2)
#' gev.tem('loc', dat = dat, plot = TRUE)
#' gev.tem('Nmean', dat = dat, p = 0.01, N=100, plot = TRUE)
#' gev.tem('Nquant', dat = dat, q = 0.5, N=100, plot = TRUE)
#' }
gev.tem <- function(param = c("loc", "scale", "shape", "quant", "Nmean", "Nquant"), dat, psi = NULL,
p = NULL, q = 0.5, N = NULL, n.psi = 50, plot = TRUE, correction = TRUE) {
if (param %in% c("VaR", "retlev"))
{
param <- "quant"
} #Compatibility following change of notation 13/07/2017
if (param == "scale" && !is.null(psi)) {
if (isTRUE(any(psi < 0))) {
stop("Invalid argument: scale parameter must be positive")
}
}
tem_out <- gev.pll(psi = psi, param = param, mod = "tem", dat = dat, N = N, p = p, q = q,
correction = correction)
if (plot) {
plot.fr(tem_out)
}
return(invisible(tem_out))
}
#' Tangent exponential model approximation for the GP distribution
#'
#' The function \code{gpd.tem} provides a tangent exponential model (TEM) approximation
#' for higher order likelihood inference for a scalar parameter for the generalized Pareto distribution. Options include
#' scale and shape parameters as well as value-at-risk (also referred to as quantiles, or return levels)
#' and expected shortfall. The function attempts to find good values for \code{psi} that will
#' cover the range of options, but the fit may fail and return an error. In such cases, the user can try to find good
#' grid of starting values and provide them to the routine.
#'
#' As of version 1.11, this function is a wrapper around \code{gpd.pll}.
#'
#' @details The interpretation for \code{m} is as follows: if there are on average \eqn{m_y} observations per year above the threshold, then \eqn{m = Tm_y} corresponds to \eqn{T}-year return level.
#'
#' @param param parameter over which to profile
#' @param threshold threshold value corresponding to the lower bound of the support or the location parameter of the generalized Pareto distribution.
#' @param psi scalar or ordered vector of values for the interest parameter. If \code{NULL} (default), a grid of values centered at the MLE is selected. If \code{psi} is of length 2 and \code{n.psi}>2, it is assumed to be the minimal and maximal values at which to evaluate the profile log likelihood.
#' @param m number of observations of interest for return levels. See \strong{Details}. Required only for \code{param = 'VaR'} or \code{param = 'ES'}.
#' @param N size of block over which to take maxima. Required only for \code{args} \code{Nmean} and \code{Nquant}.
#' @param p tail probability, equivalent to \eqn{1/m}. Required only for \code{args} \code{quant}.
#' @param q level of quantile for N-block maxima. Required only for \code{args} \code{Nquant}.
#' @param dat sample vector for the GP distribution
#' @param n.psi number of values of \code{psi} at which the likelihood is computed, if \code{psi} is not supplied (\code{NULL}). Odd values are more prone to give rise to numerical instabilities near the MLE
#' @param plot logical indicating whether \code{plot.fr} should be called upon exit
#' @param correction logical indicating whether \link{spline.corr} should be called.
#' @author Leo Belzile
#' @return an invisible object of class \code{fr} (see \code{tem} in package \code{hoa}) with elements
#' \itemize{
#' \item{\code{normal}: }{maximum likelihood estimate and standard error of the interest parameter \eqn{psi}}
#' \item{\code{par.hat}: }{maximum likelihood estimates}
#' \item{\code{par.hat.se}: }{standard errors of maximum likelihood estimates}
#' \item{\code{th.rest}: }{estimated maximum profile likelihood at (\eqn{psi}, \eqn{\hat{\lambda}})}
#' \item{\code{r}: }{values of likelihood root corresponding to \eqn{\psi}}
#' \item{\code{psi}: }{vector of interest parameter}
#' \item{\code{q}: }{vector of likelihood modifications}
#' \item{\code{rstar}: }{modified likelihood root vector}
#' \item{\code{rstar.old}: }{uncorrected modified likelihood root vector}
#' \item{\code{param}: }{parameter}
#' }
#' @export
#' @examples
#' set.seed(123)
#' dat <- rgp(n = 40, scale = 1, shape = -0.1)
#' #with plots
#' m1 <- gpd.tem(param = 'shape', n.psi = 50, dat = dat, plot = TRUE)
#' \dontrun{
#' m2 <- gpd.tem(param = 'scale', n.psi = 50, dat = dat)
#' m3 <- gpd.tem(param = 'VaR', n.psi = 50, dat = dat, m = 100)
#' #Providing psi
#' psi <- c(seq(2, 5, length = 15), seq(5, 35, length = 45))
#' m4 <- gpd.tem(param = 'ES', dat = dat, m = 100, psi = psi, correction = FALSE)
#' mev:::plot.fr(m4, which = c(2, 4))
#' plot(fr4 <- spline.corr(m4))
#' confint(m1)
#' confint(m4, parm = 2, warn = FALSE)
#' m5 <- gpd.tem(param = 'Nmean', dat = dat, N = 100, psi = psi, correction = FALSE)
#' m6 <- gpd.tem(param = 'Nquant', dat = dat, N = 100, q = 0.7, correction = FALSE)
#' }
gpd.tem <- function(dat, param = c("scale", "shape", "quant", "VaR", "ES", "Nmean", "Nquant"),
psi = NULL, m = NULL, threshold = 0, n.psi = 50, N = NULL, p = NULL, q = NULL, plot = FALSE,
correction = TRUE) {
if (param %in% c("VaR", "ES") && is.null(m)) {
stop("Parameter \"m\" missing")
}
dat <- dat - threshold
tem <- gpd.pll(psi = psi, param = param, mod = "tem", dat = dat, N = N, m = m, mle = NULL,
q = q, p = p, correction = correction)
if (plot) {
plot.fr(tem)
}
return(invisible(tem))
}
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