The following R codes implement the Example 5.13 in
Statistical Computing with R, and compare the estimate \(\hat{\theta}\) and \(\hat{\sigma}\) from stratified importance sampling to the results from importance sampling. The example illustrates that stratification can reduce the varinace of importance sampling estimator.
M <- 100000 # number of replicates
g <- function(x) {
exp(-x - log(1 + x^2)) * (x > 0) * (x < 1)
}
# importance sampling
u <- runif(M) # f3, inverse transform method
x <- -log(1 - u * (1 - exp(-1)))
fg <- g(x) / (exp(-x) / (1 - exp(-1)))
(theta.hat0 <- mean(fg))
## [1] 0.5250196
(se0 <- sd(fg))
## [1] 0.09708643
# stratified importance sampling
k <- 5 # number of strata
m <- M / k # replicates per stratum
theta.hat <- se <- numeric(k)
for (j in 1:k) {
u <- runif(m, (j - 1) / k, j / k)
x <- -log(1 - u * (1 - exp(-1)))
fg <- g(x) / (k * exp(-x) / (1 - exp(-1)))
theta.hat[j] <- mean(fg)
se[j] <- sd(fg)
}
sum(theta.hat)
## [1] 0.5248155
sum(se)
## [1] 0.01828457
The estimate \(\hat{\theta}\) is close, while the estimated standard error \(\hat{\sigma}\) is reduced from 0.0970864 to 0.0182846 with stratification.