Proving admissibility using the stepwise Bayes technique: with applications to maximum likelihood estimation

The stepwise Bayes technique is a simple but versatile method for proving admissibility of estimators under a strictly convex loss function like squared error loss. For example, when X ~ Binomial (n, [theta]), it is easy to prove that under squared error loss the MLE of [theta] is admissible using the stepwise Bayes technique. Similarly, the admissibility of the joint MLE can also be proven in cases of X ~ Multinomial and independent Binomial random variables;Furthermore, those results can be extended. Let X ~ Multinomial (n, p) where p [epsilon] [xi] = (p[subscript]0, p[subscript]1, ..., p[subscript] k): 0 ≤ p[subscript] i ≤ 1 for each i = 0, 1, ..., k and [sigma][subscript]spi=1 k p[subscript] i = 1, and p = [underline][phi]([underline][theta]) = ([phi][subscript]0([underline][theta]), [phi][subscript]1([underline][theta]), ..., [phi][subscript] k([underline][theta]))[superscript]\u27 where [underline][theta] [epsilon] [theta] = ([theta][subscript]10, [theta][subscript]11, ..., [theta][subscript] 1s[subscript]1; [theta][subscript]20, [theta][subscript]21, ..., [theta][subscript] 2s[subscript]2; ; [theta][subscript] r1, [theta][subscript] r2, ..., [theta][subscript] rs[subscript]2): 0 ≤ [theta][subscript] ij ≤ 1 for any i, j and [sigma][subscript]sp j=1 s[subscript] i [theta][subscript] ij = 1 for each i = 1, 2, ..., r. Assume [underline][phi]: [theta] → [xi] is an onto map and each [phi][subscript] i([theta]) is a monomial of [theta][subscript]10, [theta][subscript]11, ..., [theta][subscript] 1s[subscript]1; [theta][subscript]21, [theta][subscript]22, ..., [theta][subscript] 2s[subscript]2; ; [theta][subscript] r1, [theta][subscript] r2, [theta][subscript] rs[subscript] r. Then, the stepwise Bayes technique can be used to show that the MLE of [underline][theta] is admissible under squared error loss;This result is useful for proving the admissibility of maximum likelihood estimators in many areas of statistics, for example, missing data analysis, censored data analysis, log-linear models and finite population sampling problems;In contrast to the above admissibility theorem, in binomial or multinomial problems when the parameter space is restricted or truncated to a subset of the natural parameter space, the MLE may be inadmissible under squared error loss. A quite general condition for the inadmissibility of maximum likelihood estimators in such cases can be established using the stepwise Bayes technique and the complete class theorem of Brown;References. (1) Brown, L. 1981. A complete class theorem for statistical problems with finite sample spaces. Ann. of Statist. 9:1289-1300. (2) Meeden, G., and Ghosh, M. 1981. Admissibility in finite problems. Ann. Statist. 9:846-852