Parametric bootstrap

Assume we want to know the mean square error (MSE) of the sample median as a estimator of a population mean under 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))
(sigma.hat <- sd(x))
 
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))