Order statistics

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Prerequisites

Installing and loading packages

install.packages("ggplot2")

library(ggplot2) # graphics library

Problem 1: \(N(0,1)\)

• Let us pick sample size \(n=10\). Let \(X\sim N(0,1)\).
• Repeat the following \(10,000\) times:
  • Draw a random sample \(X_1,\dots,X_n\) from the normal density \(f(x)\).
    
  • For \(k\in\{1,3,5,9,10\}\), compute the \(k\)-th statistic \(y_k\).
    
• Plot the histogram of \(y_k\) for each \(k\).

f <- function(n.sim, n, k=1:n, p=pnorm, d=dnorm, r=rnorm, name, ...) {
  if (missing(name)) name <- ""
  k <- sort(unique(k))[1 <= k & k <= n]

  # Perform the simulation.
  sim <- apply(matrix(r(n.sim*n, ...), nrow=n), 2, sort)[k, , drop=FALSE]

  # Plot the requested order statistics.
  for (i in 1:length(k)) {

    # Plot the empirical distribution.
    hist(sim[i, ], freq=FALSE, 
         xlab="Value",
         sub=paste(n.sim, "iterations with sample size", n),
         main=paste(name, "order statistic", k[i]))
  }
}
#
# Set up to simulate and display the results.
#
par(mfrow=c(2,4))
n.sim <- 1e4
set.seed(17)

# Study Normal order statistics.
f(n.sim, 10, c(1,3,5,9), name="Normal(0,1)")

Problem 2: \(Gamma(1.5,5)\)

• Let us pick sample size \(n=25\). Let \(X\sim Gamma(1.5,5)\).
• Repeat the following \(10,000\) times:
  • Draw a random sample \(X_1,\dots,X_n\) from the normal density \(f(x)\).
    
  • For \(k\in\{2,4,16,24\}\) compute the \(k\)-th statistic \(y_k\).
    
• Plot the histogram of \(y_k\) for each \(k\).

#
# Set up to simulate and display the results.
#
par(mfrow=c(2,4))
n.sim <- 1e4
set.seed(17)

# Study Gamma order statistics.
# This illustrates how to use `f` for general distributions.
f(n.sim, 25, c(2, 4, 16, 24), name="Gamma(1.5,5)", 
  p=pgamma, d=dgamma, r=rgamma, shape=1.5, scale=5)

Acknowledgement

Sourced from whuber, Simulating order statistics, URL (version: 2017-04-13): see post