4. For the binomial distribution with n trials, find the minimax rule for squared error loss and normalized squared error loss, L(, a) = (a)2

4. For the binomial distribution with n trials, find the minimax rule for squared error loss and normalized squared error loss, L(θ,

a) = (θ−a)2

θ(1−θ) .

(Hint: The rules are Bayes rules or limits of Bayes rules.)

For each of these estimates, compute the frequentist risk (the expected loss with respect to f(x|θ)), and plot this risk as a function of θ (the risk plot). Use these risk plots to choose and justify an estimator when

θ is

(a) The head probability of a randomly selected coin.

(b) The failure probability for high-quality computer chips.

(c) The probability that there is life on other planets.