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#' Random Sampling of k-th Order Statistics from a Sinh-Arcsinh Distribution
#'
#'\code{order_sinharcsinh} is used to obtain a random sample of the k-th order statistic from a Sinh-Arcsinh Distribution and some associated quantities of interest.
#' @param size numeric, represents the size of the sample.
#' @param k numeric, represents the K-th smallest value from a sample.
#' @param mu numeric, represents the location parameter values.
#' @param sigma numeric, represents scale parameter values.
#' @param nu numeric, represents skewness parameter values
#' @param tau numeric, represents kurtosis tau parameter values.
#' @param n numeric, represents the size of the sample to compute the order statistic from.
#' @param p numeric, represents the 100p percentile for the distribution of the k-th order statistic. Default value is population median, p = 0.5.
#' @param alpha numeric, (1 - alpha) represents the confidence of an interval for the population percentile p of the distribution of the k-th order statistic. Default value is 0.05.
#' @param ... represents others parameters of a Sinh-Arcsinh distribution.
#'
#' @return A list with a random sample of order statistics from a Sinh-Arcsinh Distribution, the value of its join probability density function evaluated in the random sample
#' and an approximate (1 - alpha) confidence interval for the population percentile p of the distribution of the k-th order statistic.
#' @references Gentle, J, Computational Statistics, First Edition. Springer - Verlag, 2009.
#' @references Ribgy, R. and Stasinopoulos, M. (2005) Generalized Additive Models for Location Scale and Shape, Journal of the Royal Statistical Society. Applied Statistics, Series C.
#' @author Carlos Alberto Cardozo Delgado <cardozorpackages@gmail.com>.
#' @examples
#' library(orders)
#' # A sample of size 10 of the 3-th order statistics from a Sinh-Arcsinh Distribution
#' order_sinharcsinh(size=10,k=3,mu=0,sigma=1,nu=1,tau=2,n=30,p=0.5,alpha=0.02)
#' @importFrom gamlss.dist qSHASH dSHASH
#' @importFrom stats rbeta
#' @export order_sinharcsinh
order_sinharcsinh <- function(size,k,mu,sigma,nu,tau,n,p=0.5,alpha=0.05,...){
sample <- qSHASH(initial_order(size,k,n),mu,sigma,nu,tau,...)
pdf <- factorial(size)*cumprod(dSHASH(sample,mu,sigma,nu,tau,...))[size]
log_pdf <- sum(log(2:size)) + sum(log(dSHASH(sample,mu,sigma,nu,tau,...)))
if(size>5){
int_perc_est <- interval_percentile_est(p,size,sample,alpha)
return(list(sample = sample,
pdf = pdf,
log_pdf = log_pdf,
point_percentile_est = point_percentile_est(p,size,sample),
confidence_percentile_est = int_perc_est[1:2],
aprox_coverage_prob = int_perc_est[3]))
}
cat("---------------------------------------------------------------------------------------------\n")
cat("We cannot report the confidence interval. The size of the sample is less or equal than five.\n")
return(list(sample=sample,pdf=pdf))
}
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