1. Consider posterior estimation of \(\Sigma\) based on \(S|\Sigma \sim \text{Wishart}_n(\Sigma)\) and the prior distribution
    \(\Sigma^{-1}\sim \text{Wishart}_\nu( [ (\nu-p-1) \Sigma_0]^{-1})\)
    1. Find the Bayes estimator under squared-error loss.
    2. Find the Bayes estimator under Stein’s loss.
  2. Let \(Y \sim N_{n\times p}( X B ,\Sigma\otimes I)\).
    1. Derive the form of the likelihood ratio test for hypotheses of the form \(H:CB=0\) for a known matrix \(C\).
    2. Derive the form of the likelihood ratio test for hypotheses of the form \(H: D B C =0\) for known matrices \(D\) and \(C\).
    3. Consider a hypothesis of the form \(H: Gb=0\) where \(G\) is a known matrix and \(b\) is the vectorization of \(B\) (the hypothesis imples that \(b= Na\) for some matrix \(N\) and coefficients \(a\)). Discuss how you would find the MLE and perform a likelihood ratio test.