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Question 3 [3+4+3=10 points] Suppose Xilo~Gamma(y,0), i = 1,

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Question 3 [3+4+3=10 points] Suppose Xilo~Gamma(y,0), i = 1, ..., n, where y > 0 is known and 0 > 0 is unknown. Let the prior for O be given by 0-InvGamma(a,b), where a > 0 and b > 0 are known constants. Note that information in this paragraph should be used to answer the sub-questions below unless stated otherwise. (a) Write down the likelihood. Then, write down the posterior PDF of 0 up to a proportionality constant and identify the posterior distribution of 0 (what is the name of the posterior distribution and what are its parameters?). (b) Assume that ny + a > 1. Write down the posterior mean of 0. Then, derive the frequentist MSE of the Bayesian estimator given by the posterior mean (which is obtained from the posterior mean by replacing x = (x1, ..., xn) with X (X1, ... ,Xn)) assuming that the true value of 0 is 0.. Please give an explicit formula for the frequentist MSE in terms of n, y, a, b and 6. (c) Assume n = 10 and y = 1. Using the formula derived in part (b), plot the frequentist MSE of the Bayesian estimator obtained with (a, b) = (5,1) versus 0, € [0.01,2] using Matlab. In the same figure, plot the frequentist MSE of the maximum likelihood estimator ôn = 8/ny versus 0, € [0.01,2], where X = =1 X?. Give your Matlab code. = = —