8. In analyzing data from a Bin(n, ) likelihood, the MLE is MLE = Y /n, which has MSE = Ey|(MLE )2 = V ary|(MLE)

8. In analyzing data from a Bin(n, θ) likelihood, the MLE is ˆθMLE = Y /n, which has MSE = Ey|θ(ˆθMLE −θ)2 = V ary|θ(ˆθMLE) = θ(1−θ)/n. Find the MSE of the estimator ˆθBayes = (Y + 1)/(n + 2) and discuss in what contexts you would prefer it over ˆθMLE. (ˆθBayes is the estimator from equation (1.9) with μ = 1/2 and M = 2.)