# ----------------------------------------
# Authors: Andreas Alfons and Josef Holzer
# Vienna University of Technology
# ----------------------------------------
#' QQ-estimator
#'
#' Estimate the shape parameter of a Pareto distribution using a
#' quantile-quantile approach.
#'
#' 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).
#'
#' @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.
#'
#' @return The estimated shape parameter.
#'
#' @note The argument \code{x0} for the threshold (scale parameter) of the
#' Pareto distribution was introduced in version 0.2.
#'
#' @author Andreas Alfons and Josef Holzer
#'
#' @seealso \code{\link{paretoTail}}, \code{\link{fitPareto}}
#'
#' @references Kratz, M.F. and Resnick, S.I. (1996) The QQ-estimator and heavy
#' tails. \emph{Stochastic Models}, \bold{12}(4), 699--724.
#'
#' @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$db090)
#'
#' # using number of observations in tail
#' thetaQQ(eusilc$eqIncome, k = ts$k)
#'
#' # using threshold
#' thetaQQ(eusilc$eqIncome, x0 = ts$x0)
#'
#' @export
thetaQQ <- function(x, k = NULL, x0 = 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")
if(any(i <- is.na(x))) x <- x[!i] # remove missing values
x <- sort(x)
n <- length(x)
# 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))
# }
if(!haveK) { # '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))
}
## calculations
logx <- log(x[n:(n-k+1)]) # lograrithm of reversed tail
h <- -log((1:k)/(k+1))
## QQ-estimator
(k*sum(h^2) - sum(h)^2) / sum(h * (k*logx - sum(logx)))
}
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