Nothing
R/g_kiener2.R
In FatTailsR: Kiener Distributions and Fat Tails in Finance and Neuroscience
Defines functions
eskiener2
dtmqkiener2
rtmkiener2
ltmkiener2
varkiener2
qlkiener2
dlkiener2
lkiener2
dqkiener2
dpkiener2
rkiener2
qkiener2
pkiener2
dkiener2
Documented in
dkiener2
dlkiener2
dpkiener2
dqkiener2
dtmqkiener2
eskiener2
lkiener2
ltmkiener2
pkiener2
qkiener2
qlkiener2
rkiener2
rtmkiener2
varkiener2
## #' @include f_kiener1.R
#' @title Asymmetric Kiener Distribution K2
#'
#' @description
#' Density, distribution function, quantile function, random generation,
#' value-at-risk, expected shortfall (+ signed left/right tail mean)
#' and additional formulae for asymmetric Kiener distribution K2.
#'
#' @param x vector of quantiles.
#' @param q vector of quantiles.
#' @param m numeric. The median.
#' @param g numeric. The scale parameter, preferably strictly positive.
#' @param a numeric. The left tail parameter, preferably strictly positive.
#' @param w numeric. The right tail parameter, preferably strictly positive.
#' @param p vector of probabilities.
#' @param lp vector of logit of probabilities.
#' @param n number of observations. If length(n) > 1, the length is
#' taken to be the number required.
#' @param log logical. If TRUE, densities are given in log scale.
#' @param lower.tail logical. If TRUE, use p. If FALSE, use 1-p.
#' @param log.p logical. If TRUE, probabilities p are given as log(p).
#' @param signedES logical. FALSE (default) returns positive numbers for
#' left and right tails. TRUE returns negative number
#' (= \code{ltmkiener4}) for left tail and positive number
#' (= \code{rtmkiener4}) for right tail.
#'
#' @details
#' Kiener distributions use the following parameters, some of them being redundant.
#' See \code{\link{aw2k}} and \code{\link{pk2pk}} for the formulas and
#' the conversion between parameters:
#' \itemize{
#' \item{ \code{m} (mu) is the median of the distribution,. }
#' \item{ \code{g} (gamma) is the scale parameter. }
#' \item{ \code{a} (alpha) is the left tail parameter. }
#' \item{ \code{k} (kappa) is the harmonic mean of \code{a} and \code{w}
#' and describes a global tail parameter. }
#' \item{ \code{w} (omega) is the right tail parameter. }
#' \item{ \code{d} (delta) is the distortion parameter. }
#' \item{ \code{e} (epsilon) is the eccentricity parameter. }
#' }
#'
#' Kiener distributions \code{K2(m, g, a, w)} are distributions
#' with asymmetrical left
#' and right fat tails described by the parameters \code{a} (alpha) for
#' the left tail and \code{w} (omega) for the right tail. These parameters
#' correspond to the power exponent that appear in Pareto formula and
#' Karamata theorems.
#'
#' As \code{a} and \code{w} are highly correlated, the use of Kiener distributions
#' (\code{K3(..., k, d)} K4 (\code{K4(..., k, e)} is an alternate solution.
#'
#' \code{m} is the median of the distribution. \code{g} is the scale parameter
#' and is linked for any value of \code{a} and \code{w} to the density at the
#' median through the relation
#' \deqn{ g * f(x=m, g=g) = \frac{\pi}{4\sqrt{3}} \approx 0.453 }{%
#' g * f(x=m, g=g) = pi/4/sqrt(3) = 0.453 approximatively}
#'
#' When \code{a = Inf} and \code{w = Inf}, \code{g} is very close to \code{sd(x)}.
#' NOTE: In order to match this standard deviation, the value of \code{g} has
#' been updated from versions < 1.9.0 by a factor
#' \eqn{ \frac{2\pi}{\sqrt{3}}}{ 2 pi/sqrt(3) }.
#'
#' The functions \code{dkiener2347}, \code{pkiener2347} and \code{lkiener2347}
#' have no explicit forms. Due to a poor optimization algorithm, their
#' calculations in versions < 1.9 were unreliable. In versions > 1.9, a much better
#' algorithm was found and the optimization is conducted in a fast way to avoid
#' a lengthy optimization. The two extreme elements (minimum, maximum) of the
#' given \code{x} or \code{q} arguments are sent to a second order optimizer that
#' minimize the residual error of the \code{lkiener2347} function and return the
#' estimated lower and upper logit values. Then a sequence of logit values of
#' length 51 times the length of \code{x} or \code{q} is generated between these
#' lower and upper values and the corresponding quantiles are calculated with
#' the function \code{qlkiener2347}. These 51 times more numerous quantiles are
#' then compared to the original \code{x} or \code{q} arguments and the closest
#' values with their associated logit values are selected. The probabilities are then
#' calculated with the function \code{invlogit} and the densities are calculated
#' with the function \code{dlkiener2347}. The accuracy of this approach depends
#' on the sparsity of the initial \code{x} or \code{q} sequences. A 4 digits
#' accuracy can be expected, enough for most usages.
#'
#' \code{qkiener2} function is defined for p in (0, 1) by:
#' \deqn{
#' qkiener2(p,m,g,a,w) = m + \frac{\sqrt{3}}{\pi}*g*k*
#' \left(-exp\left(-\frac{logit(p)}{a} +\frac{logit(p)}{w}\right)\right)
#' }{%
#' qkiener2(p,m,g,a,w) = m + sqrt(3)/pi*g*k*(-exp(-logit(p)/a) +exp(logit(p)/w))
#' }
#' where k is the harmonic mean of the tail parameters \code{a} and \code{w}
#' calculated by \eqn{k = aw2k(a, w)}.
#'
#' \code{rkiener2} generates \code{n} random quantiles.
#'
#' In addition to the classical d, p, q, r functions, the prefixes
#' dp, dq, l, dl, ql are also provided.
#'
#' \code{dpkiener2} is the density function calculated from the probability p.
#' It is defined for p in (0, 1) by:
#' \deqn{
#' dpkiener2(p,m,g,a,w) = \frac{\pi}{\sqrt{3}}\frac{p(1-p)}{g}\frac{2}{k}
#' \left[ +\frac{1}{a}exp\left(-\frac{logit(p)}{a}\right)
#' +\frac{1}{w}exp\left( \frac{logit(p)}{w}\right) \right]^{-1}
#' }{%
#' dpkiener2(p,m,g,a,w) = pi/sqrt(3)*p*(1-p)/g*2/k/(+exp(-logit(p)/a)/a +exp(logit(p)/w)/w)
#' }
#'
#' \code{dqkiener2} is the derivate of the quantile function calculated from
#' the probability p. It is defined for p in (0, 1) by:
#' \deqn{
#' dqkiener2(p,m,g,a,w) = \frac{\sqrt{3}}{\pi}\frac{g}{p(1-p)}\frac{k}{2}
#' \left[ +\frac{1}{a}exp\left(-\frac{logit(p)}{a}\right)
#' +\frac{1}{w}exp\left( \frac{logit(p)}{w}\right) \right]
#' }{%
#' dqkiener2(p,m,g,a,w) = sqrt(3)/pi*g/p/(1-p)*k/2*( exp(-logit(p)/a)/a + exp(logit(p)/w)/w)
#' }
#'
#' \code{dlkiener2} is the density function calculated from the logit of the
#' probability lp = logit(p) defined in (-Inf, +Inf) by:
#' \deqn{
#' dlkiener2(lp,m,g,a,w) = \frac{\pi}{\sqrt{3}}\frac{p(1-p)}{g}\frac{2}{k}
#' \left[ +\frac{1}{a}exp\left(-\frac{lp}{a}\right)
#' +\frac{1}{w}exp\left( \frac{lp}{w}\right) \right]^{-1}
#' }{%
#' dlkiener2(lp,m,g,a,w) = pi/sqrt(3)*p*(1-p)/g*2/k/(+exp(-lp/a)/a +exp(lp/w)/w)
#' }
#'
#' \code{qlkiener2} is the quantile function calculated from the logit of the
#' probability. It is defined for lp in (-Inf, +Inf) by:
#' \deqn{
#' qlkiener2(lp,m,g,a,w) = m + \frac{\sqrt{3}}{\pi}*g*k*
#' \left(-exp\left(-\frac{lp}{a} +\frac{lp}{w}\right)\right)
#' }{%
#' qlkiener2(lp,m,g,a,w) = m + sqrt(3)/pi*g*k*(-exp(-lp/a) +exp(lp/w))
#' }
#'
#' \code{varkiener2} designates the Value a-risk and turns negative numbers
#' into positive numbers with the following rule:
#' \deqn{
#' varkiener2 <- if\;(p <= 0.5)\;\; (- qkiener2)\;\; else\;\; (qkiener2)
#' }{%
#' varkiener2 <- if (p <= 0.5) (- qkiener2) else (qkiener2)
#' }
#' Usual values in finance are \code{p = 0.01}, \code{p = 0.05}, \code{p = 0.95} and
#' \code{p = 0.99}. \code{lower.tail = FALSE} uses \code{1-p} rather than \code{p}.
#'
#' \code{ltmkiener2}, \code{rtmkiener2} and \code{eskiener2} are respectively the
#' left tail mean, the right tail mean and the expected shortfall of the distribution
#' (sometimes called average VaR, conditional VaR or tail VaR).
#' Left tail mean is the integrale from \code{-Inf} to \code{p} of the quantile function
#' \code{qkiener2} divided by \code{p}.
#' Right tail mean is the integrale from \code{p} to \code{+Inf} of the quantile function
#' \code{qkiener2} divided by 1-p.
#' Expected shortfall turns negative numbers into positive numbers with the following rule:
#' \deqn{
#' eskiener2 <- if\;(p <= 0.5)\;\; (- ltmkiener2)\;\; else\;\; (rtmkiener2)
#' }{%
#' eskiener2 <- if(p <= 0.5) (- ltmkiener2) else (rtmkiener2)
#' }
#' Usual values in finance are \code{p = 0.01}, \code{p = 0.025}, \code{p = 0.975} and
#' \code{p = 0.99}. \code{lower.tail = FALSE} uses \code{1-p} rather than \code{p}.
#'
#' \code{dtmqkiener2} is the difference between the left tail mean and the quantile
#' when (p <= 0.5) and the difference between the right tail mean and the quantile
#' when (p > 0.5). It is in quantile unit and is an indirect measure of the tail curvature.
#'
#' @references
#' P. Kiener, Explicit models for bilateral fat-tailed distributions and
#' applications in finance with the package FatTailsR, 8th R/Rmetrics Workshop
#' and Summer School, Paris, 27 June 2014. Download it from:
#' \url{https://www.inmodelia.com/exemples/2014-0627-Rmetrics-Kiener-en.pdf}
#'
#' P. Kiener, Fat tail analysis and package FatTailsR,
#' 9th R/Rmetrics Workshop and Summer School, Zurich, 27 June 2015.
#' Download it from:
#' \url{https://www.inmodelia.com/exemples/2015-0627-Rmetrics-Kiener-en.pdf}
#'
#' C. Acerbi, D. Tasche, Expected shortfall: a natural coherent alternative to
#' Value at Risk, 9 May 2001. Download it from:
#' \url{https://www.bis.org/bcbs/ca/acertasc.pdf}
#'
#' @seealso
#' Symmetric Kiener distribution K1 \code{\link{kiener1}},
#' asymmetric Kiener distributions K3, K4 and K7
#' \code{\link{kiener3}}, \code{\link{kiener4}}, \code{\link{kiener7}},
#' conversion functions \code{\link{aw2k}},
#' estimation function \code{\link{fitkienerX}},
#' regression function \code{\link{regkienerLX}}.
#'
#' @examples
#'
#' require(graphics)
#'
#' ### EXAMPLE 1
#' x <- seq(-5, 5, by = 0.1) ; x
#' pkiener2(x, m=0, g=1, a=2, w=5)
#' dkiener2(x, m=0, g=1, a=2, w=5)
#' lkiener2(x, m=0, g=1, a=2, w=5)
#' plot( x, pkiener2(x, m=0, g=1, a=2, w=5), las=1)
#' lines(x, pkiener1(x, m=0, g=1, k=9999))
#'
#' plot( x, dkiener2(x, m=0, g=1, a=2, w=5), las=1, type="l", lwd=2)
#' lines(x, dkiener1(x, m=0, g=1, k=9999))
#'
#' plot( x, lkiener2(x, m=0, g=1, a=2, w=5), las=1)
#' lines(x, lkiener1(x, m=0, g=1, k=9999))
#'
#'
#' p <- c(ppoints(11, a = 1), NA, NaN) ; p
#' qkiener2(p, a=2, w=5)
#' dpkiener2(p, a=2, w=5)
#' dqkiener2(p, a=2, w=5)
#'
#' varkiener2(p=0.01, a=2, w=5)
#' ltmkiener2(p=0.01, a=2, w=5)
#' eskiener2(p=0.01, a=2, w=5) # VaR and ES should be positive
#' ### END EXAMPLE 1
#'
#'
#' ### PREPARE THE GRAPHICS FOR EXAMPLES 2 AND 3
#' xx <- c(-4,-2, 0, 2, 4)
#' lty <- c( 1, 2, 3, 4, 5, 1)
#' lwd <- c( 2, 1, 1, 1, 1, 1)
#' col <- c("black","green3","cyan3","dodgerblue2","purple2","brown3")
#' lat <- c(-6.9, -4.6, -2.9, 0, 2.9, 4.6, 6.9)
#' lgt <- c("logit(0.999) = 6.9", "logit(0.99) = 4.6", "logit(0.95) = 2.9",
#' "logit(0.50) = 0", "logit(0.05) = -2.9", "logit(0.01) = -4.6",
#' "logit(0.001) = -6.9 ")
#' funleg <- function(xy, a) legend(xy, title = expression(alpha), legend = names(a),
#' lty = lty, col = col, lwd = lwd, inset = 0.02, cex = 0.8)
#' funlgt <- function(xy) legend(xy, title = "logit(p)", legend = lgt,
#' inset = 0.02, cex = 0.6)
#'
#' ### EXAMPLE 2
#' ### PROBA, DENSITY, LOGIT-PROBA, LOG-DENSITY FROM x
#' x <- seq(-5, 5, by = 0.1) ; x ; length(x)
#' a <- c(9999, 9, 5, 3, 2, 1) ; names(a) <- a
#'
#' fun1 <- function(a, x) pkiener2(x, a=a, w=5)
#' fun2 <- function(a, x) dkiener2(x, a=a, w=5)
#' fun3 <- function(a, x) lkiener2(x, a=a, w=5)
#' fun4 <- function(a, x) dkiener2(x, a=a, w=5, log=TRUE)
#'
#' mat11 <- sapply(a, fun1, x) ; head(mat11, 10)
#' mat12 <- sapply(a, fun2, x) ; head(mat12, 10)
#' mat13 <- sapply(a, fun3, x) ; head(mat13, 10)
#' mat14 <- sapply(a, fun4, x) ; head(mat14, 10)
#'
#' op <- par(mfcol = c(2,2), mar = c(2.5,3,1.5,1), las=1)
#' matplot(x, mat11, type="l", lwd=lwd, lty=lty, col=col,
#' main="pkiener2(x, m=0, g=1, a=a, w=5)", xlab="", ylab="")
#' funleg("topleft", a)
#' matplot(x, mat12, type="l", lwd=lwd, lty=lty, col=col,
#' main="dkiener2", xlab="", ylab="")
#' funleg("topleft", a)
#' matplot(x, mat13, type="l", lwd=lwd, lty=lty, col=col, yaxt="n", ylim=c(-10,10),
#' main="lkiener2", xlab="", ylab="")
#' axis(2, at=lat, las=1)
#' funleg("bottomright", a)
#' funlgt("topleft")
#' matplot(x, mat14, type="l", lwd=lwd, lty=lty, col=col, ylim=c(-8,0),
#' main="log(dkiener2)", xlab="", ylab="")
#' funleg("bottom", a)
#' par(op)
#' ### END EXAMPLE 2
#'
#'
#' ### EXAMPLE 3
#' ### QUANTILE, DIFF-QUANTILE, DENSITY, LOG-DENSITY FROM p
#' p <- ppoints(1999, a=0) ; head(p, n=10)
#' a <- c(9999, 9, 5, 3, 2, 1) ; names(a) <- a
#'
#' mat15 <- outer(p, a, \(p,a) qkiener2(p, a=a, w=5)) ; head(mat15, 10)
#' mat16 <- outer(p, a, \(p,a) dqkiener2(p, a=a, w=5)) ; head(mat16, 10)
#' mat17 <- outer(p, a, \(p,a) dpkiener2(p, a=a, w=5)) ; head(mat17, 10)
#'
#' op <- par(mfcol = c(2,2), mar = c(2.5,3,1.5,1), las=1)
#' matplot(p, mat15, type="l", xlim=c(0,1), ylim=c(-5,5),
#' lwd=lwd, lty=lty, col=col, las=1,
#' main="qkiener2(p, m=0, g=1, a=a, w=5)", xlab="", ylab="")
#' funleg("topleft", a)
#' matplot(p, mat16, type="l", xlim=c(0,1), ylim=c(0,40),
#' lwd=lwd, lty=lty, col=col, las=1,
#' main="dqkiener2", xlab="", ylab="")
#' funleg("top", a)
#' plot(NA, NA, xlim=c(-5, 5), ylim=c(0, 0.5), las=1,
#' main="qkiener2, dpkiener2", xlab="", ylab="")
#' invisible(mapply(matlines, x=as.data.frame(mat15), y=as.data.frame(mat17),
#' lwd=lwd, lty=1, col=col))
#' funleg("topright", a)
#' plot(NA, NA, xlim=c(-5, 5), ylim=c(-7, -0.5), las=1,
#' main="qkiener2, log(dpkiener2)", xlab="", ylab="")
#' invisible(mapply(matlines, x=as.data.frame(mat15), y=as.data.frame(log(mat17)),
#' lwd=lwd, lty=lty, col=col))
#' funleg("bottom", a)
#' par(op)
#' ### END EXAMPLE 3
#'
#'
#' ### EXAMPLE 4: PROCESSUS: which processus look credible?
#' ### PARAMETER a VARIES, w=4 IS CONSTANT
#' ### RUN SEED ii <- 1 THEN THE cairo_pdf CODE WITH THE 6 SEEDS
#' # cairo_pdf("K2-6x6-stocks-a.pdf")
#' # for (ii in c(1,2016,2018,2022,2023,2024)) {
#' ii <- 1
#' set.seed(ii)
#' p <- sample(ppoints(299, a=0), 299)
#' a <- c(9999, 9, 4, 3, 2, 1) ; names(a) <- a
#' mat18 <- outer(p, a, \(p,a) qkiener2(p=p, g=0.85, a=a, w=4))
#' mat19 <- apply(mat18, 2, cumsum)
#' title <- paste0(
#' "stock_", ii,
#' ": a_left = c(", paste(a[1:3], collapse = ", "), ")",
#' ", a_right = c(", paste(a[4:6], collapse = ", "), ")",
#' ", w = 4")
#' plot.ts(mat19, ann=FALSE, las=1,
#' mar.multi=c(0,3,0,1), oma.multi=c(3,0,3,0.5))
#' mtext(title, outer = TRUE, line=-1.5, font=2)
#' plot.ts(mat18, ann=FALSE, las=1,
#' mar.multi=c(0,3,0,1), oma.multi=c(3,0,3,0.5))
#' mtext(title, outer=TRUE, line=-1.5, font=2)
#' # }
#' # dev.off()
#' ### END EXAMPLE 4
#'
#'
#'
#' @name kiener2
NULL
#' @export
#' @rdname kiener2
dkiener2 <- function(x, m = 0, g = 1, a = 3.2, w = 3.2, log = FALSE) {
lp <- lkiener2(x, m, g, a, w)
v <- dlkiener2(lp, m, g, a, w)
if(log) log(v) else v
}
#' @export
#' @rdname kiener2
pkiener2 <- function(q, m = 0, g = 1, a = 3.2, w = 3.2,
lower.tail = TRUE, log.p = FALSE) {
lp <- lkiener2(x = q, m, g, a, w)
v <- if(lower.tail) invlogit(lp) else 1 - invlogit(lp)
if(log.p) log(v) else v
}
#' @export
#' @rdname kiener2
qkiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2,
lower.tail = TRUE, log.p = FALSE) {
if(log.p) p <- exp(p)
if(!lower.tail) p <- 1-p
k <- aw2k(a, w)
# since v1.9.0
v <- m + sqrt(3)/pi*g*k*(- exp(-logit(p)/a) + exp(logit(p)/w))/2
v
}
#' @export
#' @rdname kiener2
rkiener2 <- function(n, m = 0, g = 1, a = 3.2, w = 3.2) {
p <- runif(n)
v <- qkiener2(p, m, g, a, w)
v
}
#' @export
#' @rdname kiener2
dpkiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2, log = FALSE) {
k <- aw2k(a, w)
# since v1.9.0
v <- p*(1-p)*pi/sqrt(3)/g/k/( exp(-logit(p)/a)/a + exp(logit(p)/w)/w)*2
if(log) log(v) else v
}
#' @export
#' @rdname kiener2
dqkiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2, log = FALSE) {
# Compute dX/dp
k <- aw2k(a, w)
# since v1.9.0
v <- 1/p/(1-p)*sqrt(3)/pi*g*k*( exp(-logit(p)/a)/a + exp(logit(p)/w)/w)/2
if(log) log(v) else v
}
#' @export
#' @rdname kiener2
lkiener2 <- function(x, m = 0, g = 1, a = 3.2, w = 3.2) {
funnlslm2 <- function(x, m, g, a, w, lpi) {
fn <- function(lp) x - qlkiener2(lp, m, g, a, w)
opt <- minpack.lm::nls.lm(par=lpi, fn=fn)
opt$par
}
fuv <- function(u, v) which.min(abs(u-v))[1]
lk <- lkiener1(range(x), m, g, k=aw2k(a, w))
lp2 <- lk*exp(-lk*aw2d(a, w)*g^0.5)
lr2 <- funnlslm2(range(x), m, g, a, w, range(lp2))
l5 <- seq(range(lr2)[1], range(lr2)[2],
length.out=max(50001, length(x)*51))
q5 <- qlkiener2(l5, m, g, a, w)
id5 <- sapply(x, fuv, q5)
lpi <- l5[id5]
lpi
}
#' @export
#' @rdname kiener2
dlkiener2 <- function(lp, m = 0, g = 1, a = 3.2, w = 3.2, log = FALSE) {
p <- invlogit(lp)
k <- aw2k(a, w)
# since v1.9.0
v <- p*(1-p)*pi/sqrt(3)/g /k/( exp(-lp/a)/a + exp(lp/w)/w)*2
if(log) log(v) else v
}
#' @export
#' @rdname kiener2
qlkiener2 <- function(lp, m = 0, g = 1, a = 3.2, w = 3.2, lower.tail = TRUE ) {
if(!lower.tail) lp <- -lp
k <- aw2k(a, w)
# since v1.9.0
v <- m + sqrt(3)/pi*g *k *(- exp(-lp/a) + exp(lp/w))/2
v
}
#' @export
#' @rdname kiener2
varkiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2,
lower.tail = TRUE, log.p = FALSE) {
if(log.p) p <- exp(p)
if(!lower.tail) p <- 1-p
va <- p
for (i in seq_along(p)) {
va[i] <- ifelse(p[i] <= 0.5,
- qkiener2(p[i], m, g, a, w),
qkiener2(p[i], m, g, a, w))
}
va
}
#' @export
#' @rdname kiener2
ltmkiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2,
lower.tail = TRUE, log.p = FALSE) {
if (log.p) p <- exp(p)
k <- aw2k(a, w)
ltm <- if (lower.tail) {
# m+g*k/p*(
# since v1.9.2
m+sqrt(3)/pi/2*g*k/p*(
-pbeta(p, 1-1/a, 1+1/a)*beta(1-1/a, 1+1/a)
+pbeta(p, 1+1/w, 1-1/w)*beta(1+1/w, 1-1/w))
} else {
# m+g*k/p*(
# since v1.9.2
m+sqrt(3)/pi/2*g*k/p*(
-pbeta(p, 1+1/a, 1-1/a)*beta(1+1/a, 1-1/a)
+pbeta(p, 1-1/w, 1+1/w)*beta(1-1/w, 1+1/w))
}
ltm
}
#' @export
#' @rdname kiener2
rtmkiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2,
lower.tail = TRUE, log.p = FALSE) {
if (log.p) p <- exp(p)
k <- aw2k(a, w)
rtm <- if (!lower.tail) {
# m+g*k/(1-p)*(
# since v1.9.2
m+sqrt(3)/pi/2*g*k/(1-p)*(
-pbeta(1-p, 1-1/a, 1+1/a)*beta(1-1/a, 1+1/a)
+pbeta(1-p, 1+1/w, 1-1/w)*beta(1+1/w, 1-1/w))
} else {
# m+g*k/(1-p)*(
# since v1.9.2
m+sqrt(3)/pi/2*g*k/(1-p)*(
-pbeta(1-p, 1+1/a, 1-1/a)*beta(1+1/a, 1-1/a)
+pbeta(1-p, 1-1/w, 1+1/w)*beta(1-1/w, 1+1/w))
}
rtm
}
#' @export
#' @rdname kiener2
dtmqkiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2,
lower.tail = TRUE, log.p = FALSE) {
dtmq <- p
for (i in seq_along(p)) {
dtmq[i] <- ifelse(p[i] <= 0.5,
ltmkiener2(p[i], m, g, a, w, lower.tail, log.p)
- qkiener2(p[i], m, g, a, w, lower.tail, log.p),
rtmkiener2(p[i], m, g, a, w, lower.tail, log.p)
- qkiener2(p[i], m, g, a, w, lower.tail, log.p))
}
dtmq
}
#' @export
#' @rdname kiener2
eskiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2,
lower.tail = TRUE, log.p = FALSE, signedES = FALSE) {
if (log.p) p <- exp(p)
if(!lower.tail) p <- 1-p
es <- p
for (i in seq_along(p)) {
if (signedES) {
es[i] <- ifelse(p[i] <= 0.5,
ltmkiener2(p[i], m, g, a, w),
rtmkiener2(p[i], m, g, a, w))
} else {
es[i] <- ifelse(p[i] <= 0.5,
abs(ltmkiener2(p[i], m, g, a, w)),
abs(rtmkiener2(p[i], m, g, a, w)))
}
}
es
}
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Defines functions eskiener2 dtmqkiener2 rtmkiener2 ltmkiener2 varkiener2 qlkiener2 dlkiener2 lkiener2 dqkiener2 dpkiener2 rkiener2 qkiener2 pkiener2 dkiener2
Documented in dkiener2 dlkiener2 dpkiener2 dqkiener2 dtmqkiener2 eskiener2 lkiener2 ltmkiener2 pkiener2 qkiener2 qlkiener2 rkiener2 rtmkiener2 varkiener2
## #' @include f_kiener1.R
#' @title Asymmetric Kiener Distribution K2
#'
#' @description
#' Density, distribution function, quantile function, random generation,
#' value-at-risk, expected shortfall (+ signed left/right tail mean)
#' and additional formulae for asymmetric Kiener distribution K2.
#'
#' @param x vector of quantiles.
#' @param q vector of quantiles.
#' @param m numeric. The median.
#' @param g numeric. The scale parameter, preferably strictly positive.
#' @param a numeric. The left tail parameter, preferably strictly positive.
#' @param w numeric. The right tail parameter, preferably strictly positive.
#' @param p vector of probabilities.
#' @param lp vector of logit of probabilities.
#' @param n number of observations. If length(n) > 1, the length is
#' taken to be the number required.
#' @param log logical. If TRUE, densities are given in log scale.
#' @param lower.tail logical. If TRUE, use p. If FALSE, use 1-p.
#' @param log.p logical. If TRUE, probabilities p are given as log(p).
#' @param signedES logical. FALSE (default) returns positive numbers for
#' left and right tails. TRUE returns negative number
#' (= \code{ltmkiener4}) for left tail and positive number
#' (= \code{rtmkiener4}) for right tail.
#'
#' @details
#' Kiener distributions use the following parameters, some of them being redundant.
#' See \code{\link{aw2k}} and \code{\link{pk2pk}} for the formulas and
#' the conversion between parameters:
#' \itemize{
#' \item{ \code{m} (mu) is the median of the distribution,. }
#' \item{ \code{g} (gamma) is the scale parameter. }
#' \item{ \code{a} (alpha) is the left tail parameter. }
#' \item{ \code{k} (kappa) is the harmonic mean of \code{a} and \code{w}
#' and describes a global tail parameter. }
#' \item{ \code{w} (omega) is the right tail parameter. }
#' \item{ \code{d} (delta) is the distortion parameter. }
#' \item{ \code{e} (epsilon) is the eccentricity parameter. }
#' }
#'
#' Kiener distributions \code{K2(m, g, a, w)} are distributions
#' with asymmetrical left
#' and right fat tails described by the parameters \code{a} (alpha) for
#' the left tail and \code{w} (omega) for the right tail. These parameters
#' correspond to the power exponent that appear in Pareto formula and
#' Karamata theorems.
#'
#' As \code{a} and \code{w} are highly correlated, the use of Kiener distributions
#' (\code{K3(..., k, d)} K4 (\code{K4(..., k, e)} is an alternate solution.
#'
#' \code{m} is the median of the distribution. \code{g} is the scale parameter
#' and is linked for any value of \code{a} and \code{w} to the density at the
#' median through the relation
#' \deqn{ g * f(x=m, g=g) = \frac{\pi}{4\sqrt{3}} \approx 0.453 }{%
#' g * f(x=m, g=g) = pi/4/sqrt(3) = 0.453 approximatively}
#'
#' When \code{a = Inf} and \code{w = Inf}, \code{g} is very close to \code{sd(x)}.
#' NOTE: In order to match this standard deviation, the value of \code{g} has
#' been updated from versions < 1.9.0 by a factor
#' \eqn{ \frac{2\pi}{\sqrt{3}}}{ 2 pi/sqrt(3) }.
#'
#' The functions \code{dkiener2347}, \code{pkiener2347} and \code{lkiener2347}
#' have no explicit forms. Due to a poor optimization algorithm, their
#' calculations in versions < 1.9 were unreliable. In versions > 1.9, a much better
#' algorithm was found and the optimization is conducted in a fast way to avoid
#' a lengthy optimization. The two extreme elements (minimum, maximum) of the
#' given \code{x} or \code{q} arguments are sent to a second order optimizer that
#' minimize the residual error of the \code{lkiener2347} function and return the
#' estimated lower and upper logit values. Then a sequence of logit values of
#' length 51 times the length of \code{x} or \code{q} is generated between these
#' lower and upper values and the corresponding quantiles are calculated with
#' the function \code{qlkiener2347}. These 51 times more numerous quantiles are
#' then compared to the original \code{x} or \code{q} arguments and the closest
#' values with their associated logit values are selected. The probabilities are then
#' calculated with the function \code{invlogit} and the densities are calculated
#' with the function \code{dlkiener2347}. The accuracy of this approach depends
#' on the sparsity of the initial \code{x} or \code{q} sequences. A 4 digits
#' accuracy can be expected, enough for most usages.
#'
#' \code{qkiener2} function is defined for p in (0, 1) by:
#' \deqn{
#' qkiener2(p,m,g,a,w) = m + \frac{\sqrt{3}}{\pi}*g*k*
#' \left(-exp\left(-\frac{logit(p)}{a} +\frac{logit(p)}{w}\right)\right)
#' }{%
#' qkiener2(p,m,g,a,w) = m + sqrt(3)/pi*g*k*(-exp(-logit(p)/a) +exp(logit(p)/w))
#' }
#' where k is the harmonic mean of the tail parameters \code{a} and \code{w}
#' calculated by \eqn{k = aw2k(a, w)}.
#'
#' \code{rkiener2} generates \code{n} random quantiles.
#'
#' In addition to the classical d, p, q, r functions, the prefixes
#' dp, dq, l, dl, ql are also provided.
#'
#' \code{dpkiener2} is the density function calculated from the probability p.
#' It is defined for p in (0, 1) by:
#' \deqn{
#' dpkiener2(p,m,g,a,w) = \frac{\pi}{\sqrt{3}}\frac{p(1-p)}{g}\frac{2}{k}
#' \left[ +\frac{1}{a}exp\left(-\frac{logit(p)}{a}\right)
#' +\frac{1}{w}exp\left( \frac{logit(p)}{w}\right) \right]^{-1}
#' }{%
#' dpkiener2(p,m,g,a,w) = pi/sqrt(3)*p*(1-p)/g*2/k/(+exp(-logit(p)/a)/a +exp(logit(p)/w)/w)
#' }
#'
#' \code{dqkiener2} is the derivate of the quantile function calculated from
#' the probability p. It is defined for p in (0, 1) by:
#' \deqn{
#' dqkiener2(p,m,g,a,w) = \frac{\sqrt{3}}{\pi}\frac{g}{p(1-p)}\frac{k}{2}
#' \left[ +\frac{1}{a}exp\left(-\frac{logit(p)}{a}\right)
#' +\frac{1}{w}exp\left( \frac{logit(p)}{w}\right) \right]
#' }{%
#' dqkiener2(p,m,g,a,w) = sqrt(3)/pi*g/p/(1-p)*k/2*( exp(-logit(p)/a)/a + exp(logit(p)/w)/w)
#' }
#'
#' \code{dlkiener2} is the density function calculated from the logit of the
#' probability lp = logit(p) defined in (-Inf, +Inf) by:
#' \deqn{
#' dlkiener2(lp,m,g,a,w) = \frac{\pi}{\sqrt{3}}\frac{p(1-p)}{g}\frac{2}{k}
#' \left[ +\frac{1}{a}exp\left(-\frac{lp}{a}\right)
#' +\frac{1}{w}exp\left( \frac{lp}{w}\right) \right]^{-1}
#' }{%
#' dlkiener2(lp,m,g,a,w) = pi/sqrt(3)*p*(1-p)/g*2/k/(+exp(-lp/a)/a +exp(lp/w)/w)
#' }
#'
#' \code{qlkiener2} is the quantile function calculated from the logit of the
#' probability. It is defined for lp in (-Inf, +Inf) by:
#' \deqn{
#' qlkiener2(lp,m,g,a,w) = m + \frac{\sqrt{3}}{\pi}*g*k*
#' \left(-exp\left(-\frac{lp}{a} +\frac{lp}{w}\right)\right)
#' }{%
#' qlkiener2(lp,m,g,a,w) = m + sqrt(3)/pi*g*k*(-exp(-lp/a) +exp(lp/w))
#' }
#'
#' \code{varkiener2} designates the Value a-risk and turns negative numbers
#' into positive numbers with the following rule:
#' \deqn{
#' varkiener2 <- if\;(p <= 0.5)\;\; (- qkiener2)\;\; else\;\; (qkiener2)
#' }{%
#' varkiener2 <- if (p <= 0.5) (- qkiener2) else (qkiener2)
#' }
#' Usual values in finance are \code{p = 0.01}, \code{p = 0.05}, \code{p = 0.95} and
#' \code{p = 0.99}. \code{lower.tail = FALSE} uses \code{1-p} rather than \code{p}.
#'
#' \code{ltmkiener2}, \code{rtmkiener2} and \code{eskiener2} are respectively the
#' left tail mean, the right tail mean and the expected shortfall of the distribution
#' (sometimes called average VaR, conditional VaR or tail VaR).
#' Left tail mean is the integrale from \code{-Inf} to \code{p} of the quantile function
#' \code{qkiener2} divided by \code{p}.
#' Right tail mean is the integrale from \code{p} to \code{+Inf} of the quantile function
#' \code{qkiener2} divided by 1-p.
#' Expected shortfall turns negative numbers into positive numbers with the following rule:
#' \deqn{
#' eskiener2 <- if\;(p <= 0.5)\;\; (- ltmkiener2)\;\; else\;\; (rtmkiener2)
#' }{%
#' eskiener2 <- if(p <= 0.5) (- ltmkiener2) else (rtmkiener2)
#' }
#' Usual values in finance are \code{p = 0.01}, \code{p = 0.025}, \code{p = 0.975} and
#' \code{p = 0.99}. \code{lower.tail = FALSE} uses \code{1-p} rather than \code{p}.
#'
#' \code{dtmqkiener2} is the difference between the left tail mean and the quantile
#' when (p <= 0.5) and the difference between the right tail mean and the quantile
#' when (p > 0.5). It is in quantile unit and is an indirect measure of the tail curvature.
#'
#' @references
#' P. Kiener, Explicit models for bilateral fat-tailed distributions and
#' applications in finance with the package FatTailsR, 8th R/Rmetrics Workshop
#' and Summer School, Paris, 27 June 2014. Download it from:
#' \url{https://www.inmodelia.com/exemples/2014-0627-Rmetrics-Kiener-en.pdf}
#'
#' P. Kiener, Fat tail analysis and package FatTailsR,
#' 9th R/Rmetrics Workshop and Summer School, Zurich, 27 June 2015.
#' Download it from:
#' \url{https://www.inmodelia.com/exemples/2015-0627-Rmetrics-Kiener-en.pdf}
#'
#' C. Acerbi, D. Tasche, Expected shortfall: a natural coherent alternative to
#' Value at Risk, 9 May 2001. Download it from:
#' \url{https://www.bis.org/bcbs/ca/acertasc.pdf}
#'
#' @seealso
#' Symmetric Kiener distribution K1 \code{\link{kiener1}},
#' asymmetric Kiener distributions K3, K4 and K7
#' \code{\link{kiener3}}, \code{\link{kiener4}}, \code{\link{kiener7}},
#' conversion functions \code{\link{aw2k}},
#' estimation function \code{\link{fitkienerX}},
#' regression function \code{\link{regkienerLX}}.
#'
#' @examples
#'
#' require(graphics)
#'
#' ### EXAMPLE 1
#' x <- seq(-5, 5, by = 0.1) ; x
#' pkiener2(x, m=0, g=1, a=2, w=5)
#' dkiener2(x, m=0, g=1, a=2, w=5)
#' lkiener2(x, m=0, g=1, a=2, w=5)
#' plot( x, pkiener2(x, m=0, g=1, a=2, w=5), las=1)
#' lines(x, pkiener1(x, m=0, g=1, k=9999))
#'
#' plot( x, dkiener2(x, m=0, g=1, a=2, w=5), las=1, type="l", lwd=2)
#' lines(x, dkiener1(x, m=0, g=1, k=9999))
#'
#' plot( x, lkiener2(x, m=0, g=1, a=2, w=5), las=1)
#' lines(x, lkiener1(x, m=0, g=1, k=9999))
#'
#'
#' p <- c(ppoints(11, a = 1), NA, NaN) ; p
#' qkiener2(p, a=2, w=5)
#' dpkiener2(p, a=2, w=5)
#' dqkiener2(p, a=2, w=5)
#'
#' varkiener2(p=0.01, a=2, w=5)
#' ltmkiener2(p=0.01, a=2, w=5)
#' eskiener2(p=0.01, a=2, w=5) # VaR and ES should be positive
#' ### END EXAMPLE 1
#'
#'
#' ### PREPARE THE GRAPHICS FOR EXAMPLES 2 AND 3
#' xx <- c(-4,-2, 0, 2, 4)
#' lty <- c( 1, 2, 3, 4, 5, 1)
#' lwd <- c( 2, 1, 1, 1, 1, 1)
#' col <- c("black","green3","cyan3","dodgerblue2","purple2","brown3")
#' lat <- c(-6.9, -4.6, -2.9, 0, 2.9, 4.6, 6.9)
#' lgt <- c("logit(0.999) = 6.9", "logit(0.99) = 4.6", "logit(0.95) = 2.9",
#' "logit(0.50) = 0", "logit(0.05) = -2.9", "logit(0.01) = -4.6",
#' "logit(0.001) = -6.9 ")
#' funleg <- function(xy, a) legend(xy, title = expression(alpha), legend = names(a),
#' lty = lty, col = col, lwd = lwd, inset = 0.02, cex = 0.8)
#' funlgt <- function(xy) legend(xy, title = "logit(p)", legend = lgt,
#' inset = 0.02, cex = 0.6)
#'
#' ### EXAMPLE 2
#' ### PROBA, DENSITY, LOGIT-PROBA, LOG-DENSITY FROM x
#' x <- seq(-5, 5, by = 0.1) ; x ; length(x)
#' a <- c(9999, 9, 5, 3, 2, 1) ; names(a) <- a
#'
#' fun1 <- function(a, x) pkiener2(x, a=a, w=5)
#' fun2 <- function(a, x) dkiener2(x, a=a, w=5)
#' fun3 <- function(a, x) lkiener2(x, a=a, w=5)
#' fun4 <- function(a, x) dkiener2(x, a=a, w=5, log=TRUE)
#'
#' mat11 <- sapply(a, fun1, x) ; head(mat11, 10)
#' mat12 <- sapply(a, fun2, x) ; head(mat12, 10)
#' mat13 <- sapply(a, fun3, x) ; head(mat13, 10)
#' mat14 <- sapply(a, fun4, x) ; head(mat14, 10)
#'
#' op <- par(mfcol = c(2,2), mar = c(2.5,3,1.5,1), las=1)
#' matplot(x, mat11, type="l", lwd=lwd, lty=lty, col=col,
#' main="pkiener2(x, m=0, g=1, a=a, w=5)", xlab="", ylab="")
#' funleg("topleft", a)
#' matplot(x, mat12, type="l", lwd=lwd, lty=lty, col=col,
#' main="dkiener2", xlab="", ylab="")
#' funleg("topleft", a)
#' matplot(x, mat13, type="l", lwd=lwd, lty=lty, col=col, yaxt="n", ylim=c(-10,10),
#' main="lkiener2", xlab="", ylab="")
#' axis(2, at=lat, las=1)
#' funleg("bottomright", a)
#' funlgt("topleft")
#' matplot(x, mat14, type="l", lwd=lwd, lty=lty, col=col, ylim=c(-8,0),
#' main="log(dkiener2)", xlab="", ylab="")
#' funleg("bottom", a)
#' par(op)
#' ### END EXAMPLE 2
#'
#'
#' ### EXAMPLE 3
#' ### QUANTILE, DIFF-QUANTILE, DENSITY, LOG-DENSITY FROM p
#' p <- ppoints(1999, a=0) ; head(p, n=10)
#' a <- c(9999, 9, 5, 3, 2, 1) ; names(a) <- a
#'
#' mat15 <- outer(p, a, \(p,a) qkiener2(p, a=a, w=5)) ; head(mat15, 10)
#' mat16 <- outer(p, a, \(p,a) dqkiener2(p, a=a, w=5)) ; head(mat16, 10)
#' mat17 <- outer(p, a, \(p,a) dpkiener2(p, a=a, w=5)) ; head(mat17, 10)
#'
#' op <- par(mfcol = c(2,2), mar = c(2.5,3,1.5,1), las=1)
#' matplot(p, mat15, type="l", xlim=c(0,1), ylim=c(-5,5),
#' lwd=lwd, lty=lty, col=col, las=1,
#' main="qkiener2(p, m=0, g=1, a=a, w=5)", xlab="", ylab="")
#' funleg("topleft", a)
#' matplot(p, mat16, type="l", xlim=c(0,1), ylim=c(0,40),
#' lwd=lwd, lty=lty, col=col, las=1,
#' main="dqkiener2", xlab="", ylab="")
#' funleg("top", a)
#' plot(NA, NA, xlim=c(-5, 5), ylim=c(0, 0.5), las=1,
#' main="qkiener2, dpkiener2", xlab="", ylab="")
#' invisible(mapply(matlines, x=as.data.frame(mat15), y=as.data.frame(mat17),
#' lwd=lwd, lty=1, col=col))
#' funleg("topright", a)
#' plot(NA, NA, xlim=c(-5, 5), ylim=c(-7, -0.5), las=1,
#' main="qkiener2, log(dpkiener2)", xlab="", ylab="")
#' invisible(mapply(matlines, x=as.data.frame(mat15), y=as.data.frame(log(mat17)),
#' lwd=lwd, lty=lty, col=col))
#' funleg("bottom", a)
#' par(op)
#' ### END EXAMPLE 3
#'
#'
#' ### EXAMPLE 4: PROCESSUS: which processus look credible?
#' ### PARAMETER a VARIES, w=4 IS CONSTANT
#' ### RUN SEED ii <- 1 THEN THE cairo_pdf CODE WITH THE 6 SEEDS
#' # cairo_pdf("K2-6x6-stocks-a.pdf")
#' # for (ii in c(1,2016,2018,2022,2023,2024)) {
#' ii <- 1
#' set.seed(ii)
#' p <- sample(ppoints(299, a=0), 299)
#' a <- c(9999, 9, 4, 3, 2, 1) ; names(a) <- a
#' mat18 <- outer(p, a, \(p,a) qkiener2(p=p, g=0.85, a=a, w=4))
#' mat19 <- apply(mat18, 2, cumsum)
#' title <- paste0(
#' "stock_", ii,
#' ": a_left = c(", paste(a[1:3], collapse = ", "), ")",
#' ", a_right = c(", paste(a[4:6], collapse = ", "), ")",
#' ", w = 4")
#' plot.ts(mat19, ann=FALSE, las=1,
#' mar.multi=c(0,3,0,1), oma.multi=c(3,0,3,0.5))
#' mtext(title, outer = TRUE, line=-1.5, font=2)
#' plot.ts(mat18, ann=FALSE, las=1,
#' mar.multi=c(0,3,0,1), oma.multi=c(3,0,3,0.5))
#' mtext(title, outer=TRUE, line=-1.5, font=2)
#' # }
#' # dev.off()
#' ### END EXAMPLE 4
#'
#'
#'
#' @name kiener2
NULL
#' @export
#' @rdname kiener2
dkiener2 <- function(x, m = 0, g = 1, a = 3.2, w = 3.2, log = FALSE) {
lp <- lkiener2(x, m, g, a, w)
v <- dlkiener2(lp, m, g, a, w)
if(log) log(v) else v
}
#' @export
#' @rdname kiener2
pkiener2 <- function(q, m = 0, g = 1, a = 3.2, w = 3.2,
lower.tail = TRUE, log.p = FALSE) {
lp <- lkiener2(x = q, m, g, a, w)
v <- if(lower.tail) invlogit(lp) else 1 - invlogit(lp)
if(log.p) log(v) else v
}
#' @export
#' @rdname kiener2
qkiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2,
lower.tail = TRUE, log.p = FALSE) {
if(log.p) p <- exp(p)
if(!lower.tail) p <- 1-p
k <- aw2k(a, w)
# since v1.9.0
v <- m + sqrt(3)/pi*g*k*(- exp(-logit(p)/a) + exp(logit(p)/w))/2
v
}
#' @export
#' @rdname kiener2
rkiener2 <- function(n, m = 0, g = 1, a = 3.2, w = 3.2) {
p <- runif(n)
v <- qkiener2(p, m, g, a, w)
v
}
#' @export
#' @rdname kiener2
dpkiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2, log = FALSE) {
k <- aw2k(a, w)
# since v1.9.0
v <- p*(1-p)*pi/sqrt(3)/g/k/( exp(-logit(p)/a)/a + exp(logit(p)/w)/w)*2
if(log) log(v) else v
}
#' @export
#' @rdname kiener2
dqkiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2, log = FALSE) {
# Compute dX/dp
k <- aw2k(a, w)
# since v1.9.0
v <- 1/p/(1-p)*sqrt(3)/pi*g*k*( exp(-logit(p)/a)/a + exp(logit(p)/w)/w)/2
if(log) log(v) else v
}
#' @export
#' @rdname kiener2
lkiener2 <- function(x, m = 0, g = 1, a = 3.2, w = 3.2) {
funnlslm2 <- function(x, m, g, a, w, lpi) {
fn <- function(lp) x - qlkiener2(lp, m, g, a, w)
opt <- minpack.lm::nls.lm(par=lpi, fn=fn)
opt$par
}
fuv <- function(u, v) which.min(abs(u-v))[1]
lk <- lkiener1(range(x), m, g, k=aw2k(a, w))
lp2 <- lk*exp(-lk*aw2d(a, w)*g^0.5)
lr2 <- funnlslm2(range(x), m, g, a, w, range(lp2))
l5 <- seq(range(lr2)[1], range(lr2)[2],
length.out=max(50001, length(x)*51))
q5 <- qlkiener2(l5, m, g, a, w)
id5 <- sapply(x, fuv, q5)
lpi <- l5[id5]
lpi
}
#' @export
#' @rdname kiener2
dlkiener2 <- function(lp, m = 0, g = 1, a = 3.2, w = 3.2, log = FALSE) {
p <- invlogit(lp)
k <- aw2k(a, w)
# since v1.9.0
v <- p*(1-p)*pi/sqrt(3)/g /k/( exp(-lp/a)/a + exp(lp/w)/w)*2
if(log) log(v) else v
}
#' @export
#' @rdname kiener2
qlkiener2 <- function(lp, m = 0, g = 1, a = 3.2, w = 3.2, lower.tail = TRUE ) {
if(!lower.tail) lp <- -lp
k <- aw2k(a, w)
# since v1.9.0
v <- m + sqrt(3)/pi*g *k *(- exp(-lp/a) + exp(lp/w))/2
v
}
#' @export
#' @rdname kiener2
varkiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2,
lower.tail = TRUE, log.p = FALSE) {
if(log.p) p <- exp(p)
if(!lower.tail) p <- 1-p
va <- p
for (i in seq_along(p)) {
va[i] <- ifelse(p[i] <= 0.5,
- qkiener2(p[i], m, g, a, w),
qkiener2(p[i], m, g, a, w))
}
va
}
#' @export
#' @rdname kiener2
ltmkiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2,
lower.tail = TRUE, log.p = FALSE) {
if (log.p) p <- exp(p)
k <- aw2k(a, w)
ltm <- if (lower.tail) {
# m+g*k/p*(
# since v1.9.2
m+sqrt(3)/pi/2*g*k/p*(
-pbeta(p, 1-1/a, 1+1/a)*beta(1-1/a, 1+1/a)
+pbeta(p, 1+1/w, 1-1/w)*beta(1+1/w, 1-1/w))
} else {
# m+g*k/p*(
# since v1.9.2
m+sqrt(3)/pi/2*g*k/p*(
-pbeta(p, 1+1/a, 1-1/a)*beta(1+1/a, 1-1/a)
+pbeta(p, 1-1/w, 1+1/w)*beta(1-1/w, 1+1/w))
}
ltm
}
#' @export
#' @rdname kiener2
rtmkiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2,
lower.tail = TRUE, log.p = FALSE) {
if (log.p) p <- exp(p)
k <- aw2k(a, w)
rtm <- if (!lower.tail) {
# m+g*k/(1-p)*(
# since v1.9.2
m+sqrt(3)/pi/2*g*k/(1-p)*(
-pbeta(1-p, 1-1/a, 1+1/a)*beta(1-1/a, 1+1/a)
+pbeta(1-p, 1+1/w, 1-1/w)*beta(1+1/w, 1-1/w))
} else {
# m+g*k/(1-p)*(
# since v1.9.2
m+sqrt(3)/pi/2*g*k/(1-p)*(
-pbeta(1-p, 1+1/a, 1-1/a)*beta(1+1/a, 1-1/a)
+pbeta(1-p, 1-1/w, 1+1/w)*beta(1-1/w, 1+1/w))
}
rtm
}
#' @export
#' @rdname kiener2
dtmqkiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2,
lower.tail = TRUE, log.p = FALSE) {
dtmq <- p
for (i in seq_along(p)) {
dtmq[i] <- ifelse(p[i] <= 0.5,
ltmkiener2(p[i], m, g, a, w, lower.tail, log.p)
- qkiener2(p[i], m, g, a, w, lower.tail, log.p),
rtmkiener2(p[i], m, g, a, w, lower.tail, log.p)
- qkiener2(p[i], m, g, a, w, lower.tail, log.p))
}
dtmq
}
#' @export
#' @rdname kiener2
eskiener2 <- function(p, m = 0, g = 1, a = 3.2, w = 3.2,
lower.tail = TRUE, log.p = FALSE, signedES = FALSE) {
if (log.p) p <- exp(p)
if(!lower.tail) p <- 1-p
es <- p
for (i in seq_along(p)) {
if (signedES) {
es[i] <- ifelse(p[i] <= 0.5,
ltmkiener2(p[i], m, g, a, w),
rtmkiener2(p[i], m, g, a, w))
} else {
es[i] <- ifelse(p[i] <= 0.5,
abs(ltmkiener2(p[i], m, g, a, w)),
abs(rtmkiener2(p[i], m, g, a, w)))
}
}
es
}
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