Admissibility of unbiased procedures.

(i) Under the assumptions of Problem 1.10, if among the unbiased procedures there exists one with uniformly minimum risk, it is admissible. (ii) That in general an unbiased procedure with uniformly minimum risk need not be admissible is seen by the following example. Let X have a Poisson distribution truncated at 0, so that Pθ{X = x} = θxe−θ/[x!(1 − e−θ)] for x = 1, 2,.... For estimating

γ(θ) = e−θ with loss function L(θ, d)=(d−e−θ)

2, there exists a unique unbiased estimate, and it is not admissible.

[(ii): The unique unbiased estimate δ0(x)=(−1)x+1 is dominated by δ1(x)=0 or 1 as x is even or odd.]