In Example 6.8-2, take n = 30, ? = 15, and ? = 5.(a) Using

QUESTION:

In Example 6.8-2, take n = 30, \(\alpha=15\), and \(\beta=5\).

(a) Using the squared error loss, compute the expected loss (risk function) associated with the Bayes estimator w(Y).

(b) The risk function associated with the usual estimator Y/n is, of course, \(\theta(1-\theta) / 30\). Find those values of \(\theta\) for which the risk function in part (a) is less than \(\theta(1-\theta) / 30\). In particular, if the prior mean \(\alpha /(\alpha+\beta)=3 / 4\) is a reasonable guess, then the risk function in part (a) is the better of the two (i.e., is smaller in a neighborhood of \(\theta=3 / 4\)) for what values of \(\theta\)?