ABSTRACT
S is unknown. Let S-1=R. Suppose that the joint prior density ? (?) of (µ, R) is given by (5.31), which implies that the conditional prior of µ given R is of freedom and parameter H with v=2a-p>p. From this it follows that Np(a, S
- 1b) and the marginal prior of R is a Wishart Wp(a, H), with a degrees the posterior joint density ? (?|X=x) of (µ, R) is the product of the conditional posterior density II(µ|R=r, X=x) given R=r, X=x and the marginal posterior density ? (r|X=x) of R given X=x where
with Thus the Bayes estimator of µ is given by
For the loss (5.41) its risk is p(N+b)- 1. Hence, taking the expectation with respect to ? (?), we obtain
Thus is for every Since has constant risk p/N we conclude that is minimax when S is unknown.
