The admissibility of some generalized and stepwise Bayes estimators
Consider an estimation problem in the one parameter exponential family of distributions under squared error loss. Das Gupta and Sinha (1984) and Meeden and Ghosh gave, using an approach given in Brown and Hwang (1982) which is in turn based on Blyth's (1951) method, two different sets of sufficient conditions for admissibility of generalized Bayes estimators of an arbitrary parametric function. These two sets of sufficient conditions are discussed and compared;Also, using Karlin's technique, sufficient conditions are given for generalized Bayes estimators to be admissible under squared error loss for estimating an arbitrary nonnegative, differentiable, strictly increasing or decreasing parametric function in one parameter nonregular families. Some examples are subsequently given;Finally, we consider estimating an arbitrary parametric function in the case when the parameter and sample spaces are countable and the decision space is arbitrary. Using the notions of a stepwise Bayes procedure and finite admissibility, a theorem is proved which shows that every finitely admissible estimator is unique stepwise Bayes. Under an additional assumption, it is shown that the converse is true as well. The first result is also extended to the case when the parameter and sample spaces are arbitrary, i.e., not necessarily countable. In a special setting in which the parameters, sample, and decision spaces are all countable, it is shown that the class of all admissible estimators is exactly the same as that of all finitely admissible estimators.