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))

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