Chapter 6, Problem 34SE

The mean squared error of an estimator $\hat{\theta }$ is MSE ( $\hat{\theta }$) = E( $\hat{\theta }$ − $\hat{\theta }$)². If $\hat{\theta }$ is unbiased, then MSE( $\hat{\theta }$) = V( $\hat{\theta }$), but in general MSE( $\hat{\theta }$) = V( $\hat{\theta }$) + (bias)². Consider the estimator $\widehat{{\sigma }^2}$ = KS², where S² = sample variance. What value of K minimizes the mean squared error of this estimator when the population distribution is normal? [Hint: It can be shown that

$E[{(S^2)}^2]\text{ = (}n\text{ + 1)}{\sigma }^4/(n\text{ - 1})$

In general, it is difficult to find $\hat{\theta }$ to minimize MSE( $\hat{\theta }$), which is why we look only at unbiased estimators and minimize V( $\hat{\theta }$).]