8.1 Determine the limit distribution of the Bayes estimator corresponding to squared error loss, and verify that it is asymptotically efficient, in each of the following cases:

(a) The observations X1,...,Xn are iid N(θ,σ2), with σ known, and the estimand is

θ. The prior distribution for is a conjugate normal distribution, say N(µ, b2).

(See Example 4.2.2.)

(b) The observations Yi have the gamma distribution H(γ , 1/τ ), the estimand is 1/τ , and τ has the conjugate prior density H(g, α).

(c) The observations and prior are as in Problem 4.1.9 and the estimand is λ.

(d) The observations Yi have the negative binomial distribution (4.3), p has the prior density B

(a, b), and the estimand is

(a) p and

(b) 1/b.