# ----------------------------------------
# Authors: Andreas Alfons and Josef Holzer
# Vienna University of Technology
# ----------------------------------------
#' Integrated squared error (ISE) estimator
#'
#' The integrated squared error (ISE) estimator estimates the shape parameter of
#' a Pareto distribution based on the relative excesses of observations above a
#' certain threshold.
#'
#' The arguments \code{k} and \code{x0} of course correspond with each other.
#' If \code{k} is supplied, the threshold \code{x0} is estimated with the \eqn{n
#' - k} largest value in \code{x}, where \eqn{n} is the number of observations.
#' On the other hand, if the threshold \code{x0} is supplied, \code{k} is given
#' by the number of observations in \code{x} larger than \code{x0}. Therefore,
#' either \code{k} or \code{x0} needs to be supplied. If both are supplied,
#' only \code{k} is used (mainly for back compatibility).
#'
#' The ISE estimator minimizes the integrated squared error (ISE) criterion with
#' a complete density model. The minimization is carried out using %
#' \code{\link[stats]{nlm}}. By default, the starting value is obtained % with
#' the Hill estimator (see \code{\link{thetaHill}}).
#' \code{\link[stats]{optimize}}.
#'
#' @param x a numeric vector.
#' @param k the number of observations in the upper tail to which the Pareto
#' distribution is fitted.
#' @param x0 the threshold (scale parameter) above which the Pareto distribution
#' is fitted.
#' @param w an optional numeric vector giving sample weights.
#' @param \dots additional arguments to be passed to
#' \code{\link[stats]{optimize}} (see \dQuote{Details}).
#'
#' @return The estimated shape parameter.
#'
#' @note The arguments \code{x0} for the threshold (scale parameter) of the
#' Pareto distribution and \code{w} for sample weights were introduced in
#' version 0.2.
#'
#' @author Andreas Alfons and Josef Holzer
#'
#' @seealso \code{\link{paretoTail}}, \code{\link{fitPareto}},
#' \code{\link{thetaPDC}}, \code{\link{thetaHill}}
#'
#' @references
#' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators
#' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of
#' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15}
#'
#' A. Alfons, M. Templ, P. Filzmoser (2013) Robust estimation of economic
#' indicators from survey samples based on Pareto tail modeling. \emph{Journal
#' of the Royal Statistical Society, Series C}, \bold{62}(2), 271--286.
#'
#' Vandewalle, B., Beirlant, J., Christmann, A., and Hubert, M.
#' (2007) A robust estimator for the tail index of Pareto-type
#' distributions. \emph{Computational Statistics & Data Analysis},
#' \bold{51}(12), 6252--6268.
#'
#' @keywords manip
#'
#' @examples
#' data(eusilc)
#' # equivalized disposable income is equal for each household
#' # member, therefore only one household member is taken
#' eusilc <- eusilc[!duplicated(eusilc$db030),]
#'
#' # estimate threshold
#' ts <- paretoScale(eusilc$eqIncome, w = eusilc$db090)
#'
#' # using number of observations in tail
#' thetaISE(eusilc$eqIncome, k = ts$k, w = eusilc$db090)
#'
#' # using threshold
#' thetaISE(eusilc$eqIncome, x0 = ts$x0, w = eusilc$db090)
#'
#' @export
thetaISE <- function(x, k = NULL, x0 = NULL, w = NULL, ...) {
## initializations
if(!is.numeric(x) || length(x) == 0) stop("'x' must be a numeric vector")
haveK <- !is.null(k)
if(haveK) { # if 'k' is supplied, it is always used
if(!is.numeric(k) || length(k) == 0 || k[1] < 1) {
stop("'k' must be a positive integer")
} else k <- k[1]
} else if(!is.null(x0)) { # otherwise 'x0' (threshold) is used
if(!is.numeric(x0) || length(x0) == 0) stop("'x0' must be numeric")
else x0 <- x0[1]
} else stop("either 'k' or 'x0' must be supplied")
haveW <- !is.null(w)
if(haveW) { # sample weights are supplied
if(!is.numeric(w) || length(w) != length(x)) {
stop("'w' must be numeric vector of the same length as 'x'")
}
if(any(w < 0)) stop("negative weights in 'w'")
if(any(i <- is.na(x))) { # remove missing values
x <- x[!i]
w <- w[!i]
}
# sort values and sample weights
order <- order(x)
x <- x[order]
w <- w[order]
} else { # no sample weights
if(any(i <- is.na(x))) x <- x[!i] # remove missing values
x <- sort(x) # sort values
}
.thetaISE(x, k, x0, w, ...)
}
# internal function that assumes that data are ok and sorted
.thetaISE <- function(x, k = NULL, x0 = NULL, w = NULL,
tol = .Machine$double.eps^0.25, ...) {
n <- length(x) # number of observations
haveK <- !is.null(k)
haveW <- !is.null(w)
if(haveK) { # 'k' is supplied, threshold is determined
if(k >= n) stop("'k' must be smaller than the number of observed values")
x0 <- x[n-k] # threshold (scale parameter)
} else { # 'k' is not supplied, it is determined using threshold
# values are already sorted
if(x0 >= x[n]) stop("'x0' must be smaller than the maximum of 'x'")
k <- length(which(x > x0))
}
## computations
y <- x[(n-k+1):n]/x0 # relative excesses
if(haveW) {
wTail <- w[(n-k+1):n]
## weighted integrated squared error distance criterion
# w ... sample weights
ISE <- function(theta, y, w) {
f <- theta*y^(-1-theta)
weighted.mean(theta^2/(2*theta+1) - 2*f, w)
}
} else {
wTail <- NULL
## integrated squared error distance criterion
# w ... sample weights (not needed here, only available to have the
# same function definition)
ISE <- function(theta, y, w) {
f <- theta*y^(-1-theta)
mean(theta^2/(2*theta+1) - 2*f)
}
}
## optimize
localOptimize <- function(f, interval = NULL, tol, ...) {
if(is.null(interval)) {
p <- if(haveK) .thetaHill(x, k, w=w) else .thetaHill(x, x0=x0, w=w)
interval <- c(0 + tol, 3 * p) # default interval
}
optimize(f, interval, ...)
}
localOptimize(ISE, y=y, w=wTail, tol=tol, ...)$minimum
}
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