ABSTRACT
A second interpretation of James-Stein estimator is a Bayes estimator given in (5.45-5.46) where X is distributed as Np(µ, I) and the prior density of µ is Np(0, bI) with b unknown. The Bayes estimator of µ in this setup is
To estimate b we proceed as follows. Since X-µ given µ is Np(0, (1+b)I), which
implies that (1+b)- 1X'X is distributed as Hence E((1+b)/X'X)= 1/p-2, provided p>2. Thus a reasonable estimate of (1+b)- 1 is (p-2)/X'X. So the Bayes estimator of µ is
which is the James-Stein estimator of µ. If cov(X)=s2I, the Bayes estimator of µ is (1-((p-2)s2/X'X)X.
