5.7 Faith (1978) considered the hierarchical model X| N( , I ), |t N...

5.7 Faith (1978) considered the hierarchical model X|θ ∼ N(θ , I ),

θ |t ∼ N



0, 1 t

I



, t ∼ Gamma

(a, b), that is,

π(t) =

1 H(a)ba t a−1 e−t/b.

(a) Show that the marginal prior for θ , unconditional on t, is

π(θ ) ∝ (2/b + |θ |

2

)

−(a+r/2), a multivariate Student’s t-distribution.

(b) Show that a ≤ −1 is a sufficient condition for

i

∂2π(θ )

∂θ2 i

≥ 0 and, hence, is a sufficient condition for the minimaxity of the Bayes estimator against squared error loss.

(c) Show, more generally, that the Bayes estimator against squared error loss is minimax if a ≤ (r − 4)/2 and a ≤ 1/b + 3.