Hypothesis testing in multiparameter exponential families

The classical problem of testing a simple null hypothesis (with nuisance parameters) in a k-parameter exponential family is reviewed.

For families with continuous support (e.g., multivariate normal distributions) complete class theorems of A. Birnbaum, Matthes/Truax, and Eaton provide necessary conditions for admissibility: convexity (and monotonicity) of the acceptance region for general (resp., one-sided) alternatives, while Stein's method of distant alternatives gives sufficient conditions for admissibility.

Matthes/Truax and Ledwina presented analogous complete class theorems for discrete exponential families with finite support (e.g., multinomial distributions), but their results contain a small but significant gap. Fortunately this is easily corrected, and in the process leads to new results for discrete exponential families with infinite but sigma-finite support, e.g., Poisson sampling in contingency tables, discrete-time stochastic processes with discrete state spaces.

This work is inspired by and dedicated to Charles M. Stein, esteemed teacher, mentor, and humanitarian.