1.2 J allies-Stein estimator and empirical Bayes Suppose X ~ N01,, 1') is a ddimensional multivariate normal random variable. :1 2 3. A natural estimate and also MLE of a is simply ft = X. James-Stein estimator can be deriVed in the following Way: We put prior it ~ N\"), VQI). (a) Show under this prior, the marginal distribution of X becomes N0], (1 + V2)\" and marginal distribution of ||X||E ~ (1 + 1/2))(3. (b) Use Bayes formula to show the posterior mean of ,u given X is (1 11%)X. (c) The James-Stein estimator can be derived by replacing \"1\"; by g, namely, 2 d2 IIXIIE :1\" = (1 )X. Formally justify Why we shall replace Hlpg' by g. Where does at - 2 come from? 2 (d) Show that for any xed p, ]E[|| pug] = d. This is the mean square error of MLE. (c) For any xed in and any function m) : Rd + R (satisfying certain regularity condition), using the density function and integration by part, to show that MaeLEM = Covumxk). This is known as Stein's lemma