Week 10 - Discussion

The Rayleigh density [156, (18.76)] is

\[f(x) = \frac{x}{\sigma^2}e^{-x^2/(2\sigma^2)},~~x \geq0,~~ \sigma > 0\] Implement a function to generate samples from a Rayleigh distribution using antithetic variables. What is the percent reduction in variance of \(\frac{X+X^`}{2}\) compared with \(\frac{X_1+X_2}{2}\) for independent \(X_1\), \(X_2\)?

library(tidyverse)
library(scales)

set.seed(1234)

rayleigh_dist <- function(x, sigma){
                 return(sigma * sqrt(-2 * log(x))) #(x/sigma^2) * exp(-x^2)/(2*sigma^2)) # https://en.wikipedia.org/wiki/Rayleigh_distribution - Generating random variates
                 }

antithetic_rayleigh <- function(n, sig){
              list_ <- list(u = runif(n)) %>%
                          c(., list(v = 1 - .$u,
                                    v2 = runif(n))) %>%
                          c(., list(ray_dist_u    = rayleigh_dist(.$u, sig),
                                    ray_dist_v    = rayleigh_dist(.$v, sig),
                                    ray_dist_v2   = rayleigh_dist(.$v2, sig))) %>%
                          c(., list(ray_dist_var      = var((.$ray_dist_u + .$ray_dist_v2)/2),
                                    ray_dist_anti_var = var((.$ray_dist_u + .$ray_dist_v)/2))) 
              return(list_)
}
results <- antithetic_rayleigh(1000, 3)

percent((results$ray_dist_anti_var - results$ray_dist_var ) / results$ray_dist_var)

## [1] "-93.9%"

There is a profound effect on the variance reduction when there is a negative correlation compared to independence.

Rizzo; Maria L.. Statistical Computing with R (Chapman & Hall/CRC The R Series).