Parametric bootstrap

Assume we want to know the mean square error (MSE) of the sample median as a estimator of a population meanunder normality. As you know, this is not a trivial problem. We may take advantage of the Bootstrap method and solve it by means of simulation.

This way, for \(b=1, \ldots, B\), we generate \(X_{b1}, \ldots, X_{bn} \sim N(\hat{\mu}, \hat{\sigma}^2)\). Then, we compute the sample median \(\tilde{X}_b\) for each sample in the bootstrap. Finally, an estimator of the MSE is given by

\[\widehat{MSE} = B^{-1} \sum_{b=1}^B(\tilde{X}_b - \hat{\mu})^2\]

In R, the simulation should look like this for a sample size of ten units:

n <- 10
x <- rnorm(n, 10, 1)
(mu.hat <- mean(x))

[1] 10.11619

(sigma.hat <- sd(x))

[1] 1.148321

boot.MSE.median <- function(B, mu.hat, sigma.hat){
  x.s <- rnorm(n, mu.hat, sigma.hat)
  SE <- (median(x.s) - mean(mu.hat)) ^ 2
}

B <- 500
boot.SE <- replicate(B, boot.MSE.median(B, mu.hat, sigma.hat))
(MSE.hat <- mean(boot.SE))

[1] 0.1865214