Question

Obtain a Monte Carlo estimate of

dr =e V27

by importance sampling using R.

Hint: Use the pdf of Pareto(1,1) for the importance function.

Answer 1

The Pareto distribution is $f(x)=\frac{1}{x^2};x>1$ . This is the importance function.

Let $g(x)=\frac{x^2}{\sqrt{2\pi }}e^{-x^2/2}$ . We need to evaluate $\int_{1}^{\infty }g(x)dx$ . Using the above importance sampling function,

$\int_{1}^{\infty }g(x)dx=\int_{1}^{\infty }g(x)\frac{f(x)}{f(x)}dx\\ \int_{1}^{\infty }g(x)dx=\int_{1}^{\infty }\frac{g(x)}{f(x)}f(x)dx\\ \int_{1}^{\infty }g(x)dx=E_f\left ( \frac{g(X)}{f(X)} \right )$

The R code for finding the value of $\int_{1}^{\infty }\frac{x^2}{\sqrt{2\pi }}e^{-x^2/2}dx$ is given below.

library(VGAM)
set.seed(1567)
g <- function(x){
x^2*exp(-x^2/2)/sqrt(2*pi)
}
imp_mc <- function(f, B){
x <- rpareto(B, scale=1,shape=1)
gg <- g(x) / dpareto(x, scale=1,shape=1)
return(gg)
}
B <- 100000
I <- imp_mc(g, B)
mean(I)

The simulated value is ${\color{Blue} \int_{1}^{\infty }\frac{x^2}{\sqrt{2\pi }}e^{-x^2/2}dx=0.4018581}$ . The exact value is 0.400626.