R/order_sm.R

In orders: Sampling from k-th Order Statistics of New Families of Distributions

#' Random Sampling of k-th Order Statistics from a Singh-Maddala Distribution
#'
#'\code{order_sm} is used to obtain a random sample of the k-th order statistic from a Singh-Maddala distribution and some associated quantities of interest.
#' @param size numeric, represents the size of the sample.
#' @param shape1 numeric, represents a first shape parameter value. Must be strictly positive.
#' @param shape2 numeric, represents a second shape parameter value. Must be strictly positive.
#' @param scale numeric, represents scale parameter values. Must be strictly positive.
#' @param k numeric, represents the K-th smallest value from a sample.
#' @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 Singh-Maddala distribution.
#' @return A list with a random sample of order statistics from a Singh-Maddala 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 Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.
#' @author Carlos Alberto Cardozo Delgado <cardozorpackages@gmail.com>.
#' @examples
#' library(orders)
#' # A sample of size 10 of the 3-th order statistics from a Singh-Maddala Distribution
#' order_sm(size=10,shape1=1,shape2=2,scale=1,k=3,n=50,p=0.5,alpha=0.02)
#' @importFrom VGAM qsinmad dsinmad
#' @importFrom stats rbeta
#' @export order_sm

order_sm <- function(size,k,shape1,shape2,scale,n,p=0.5,alpha=0.05,...){
  sample  <- qsinmad(initial_order(size,k,n),shape1,shape2,scale,...)
  pdf     <- factorial(size)*cumprod(dsinmad(sample,shape1,shape2,scale,...))[size]
  log_pdf     <- sum(log(2:size)) + sum(log(dsinmad(sample,shape1,shape2,scale,...)))
  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))
}