7.17 (Empirical Bayes estimation in a general case). A general version of the hierarchical models of Examples 7.7 and 7.8 is X|ξ ∼ Ns(ξ, σ2 I ),

ξ|β ∼ Ns(Zβ, τ 2 I )

where σ2 and Zs×r, of rank r, are known and τ 2 and βr×1 are unknown. Under this model show that:

(a) The Bayes estimator of ξ , under squared error loss, is E(ξ|x, β) = σ2

σ2 + τ 2 zβ + τ 2

σ2 + τ 2 x.

(b) Marginally, the distribution of X|β is X|β ∼ Ns(Zβ, (σ2 + τ 2)I ).

(c) Under the marginal distribution in part (b),