Let X1 Xn be a random sample from a ntheta sigma2 population

Let X_1,..., X_n be a random sample from a n(theta, sigma^2) population, sigma ^2 known. Consider estimating theta using squared error loss. Let pi(theta) be a n(mu, tau ^2) prior distribution on theta and let delta ^pi be the Bayes estimator of theta. Verify the following formulas for the risk function and Bayes risk. For any constants a and b, the estimator delta(x) = aX- + b has risk function Let eta = sigma^2/(n tau^2 + sigma ^2). The risk function for the Bayes estimator is The Bayes risk for the Bayes estimator is B(pi, delta ^pi) = r^2 eta.

Solution