7.25 Theorem 7.17 also applies to the Poisson() case, where Johnstone (1984) obtained the following characterization of

7.25 Theorem 7.17 also applies to the Poisson(λ) case, where Johnstone (1984) obtained the following characterization of admissible estimators for the loss L(λ, δ) =
r i=1(λi −

δi)

2/λi.

A generalized Bayes estimator of the form δ(x) = [1 − h(xi)]x is

(i) inadmissible if there exists ε > 0 and M < ∞ such that h(xi) < r − 1 − ε

xi for xi > M,

(ii) admissible if h(xi)(xi)

1/2 is bounded and there exits M < ∞ such that h(xi) ≥ r − 1

xi for xi > M.

(a) Use Johnstone’s characterization of admissible Poisson estimators (Example 7.22)

to find an admissible Clevenson-Zidek estimator (6.31).

(b) Determine conditions under which the estimator is both admissible and minimax.