Write a program in \(\mathrm{R}\) that will simulate \(b=1000\) samples of size \(n\) from a \(\mathrm{T}(u)\) distribution. For each sample the compute the sample mean, the sample mean with \(5 \%\) trimming, the sample mean with \(10 \%\) trimming, and the sample median. Estimate the mean squared error of estimating the population mean for each of these estimators over the \(b\) samples. Use your program to obtain the mean square estimates when \(n=25,50\), and 100 with \(u=3,4,5,10\), and 25 .

a. Informally compare the results of these simulations. Does the sample median appear to be more efficient than the sample mean as indicated by the asymptotic relative efficiency when \(u\) equals three and four? Does the trend reverse itself when \(u\) becomes larger? How do the trimmed mean methods compare to the sample mean and the sample median?

b. Now formally compare the four estimators using an analysis of variance with a randomized complete block design where the treatments are taken to be the estimators, the blocks are taken to be the sample sizes, and the observed mean squared errors are taken to be the observations. How do the results of this analysis compare to the results observed above?