Question

4x3dx. 6. Consider estimating θ (a) Construct standard simulation estimator (b) Construct importance sampling estimator (choose appropriate pro- posal density) (c) Compare the variances of the above two estimators and illustrate importance sampling estimator achieving variance reduction.

Answer 1

a) A standard simulation estimator of the integral is implemented in R below:
set.seed(1567)
f <- function(x){
4*x^3
}
std_mc <- function(f, B){
x <- runif(B)
ff <- f(x)
return(ff)
}
B <- 100000
I <- std_mc(f,B)
mean(I)
var(I)

b) An importance sampling estimator based exponential distribution (weight function)  $f(x)=\lambda e^{-\lambda x};x>0$ is given below.

set.seed(1567)
f <- function(x){
4*x^3
}

imp_mc <- function(lambda, f, B){
x <- rexp(B, lambda)
ff <- pexp(1,lambda)*f(x[x<1.0]) / dexp(x[x<1.0], lambda)
return(ff)
}
lambda <- 0.045
B <- 100000
I <- imp_mc(lambda, f, B)
mean(I)
var(I)

c) The variance in first case is
1.273779
and variance in second case (importance sampling) is
1,259844
thus achieving variance reduction.