Show that for the multivariate normal model with \(n\) observations, the MLE of \((\mu, \Sigma)\) is \((\bar{y}, S/n)\), whereas for the Wishart model with \(n-1\) degrees of freedom, the MLE of \(\Sigma\) is \(S/(n-1)\). Show that for the normal model with \(n\) observations where the mean \(\mu\) is known to be zero, the MLE of \(\Sigma\) is \(S/n\).
Prove that \(S/n\) is the best \(GL\)-equivariant estimator under Stein’s loss.
Prove Theorem 26 in the notes, characterizing the orthogonally equivariant estimators.
Prove Theorem 27 in the notes, describing the risk of orthogonally equivariant estimators.