Admissible and Minimax Integer-Valued Estimators of an Inter-Valued Parameter

Abstract

The decision problem considered here is that of deciding which element of a finite parametric family of probability distributions $p(x, \mu)$ represents the true distribution of the statistic $X$. It is assumed that $p(x, \mu)$ satisfies certain regularity conditions which essentially require that the parameter $\mu$ be integer-valued with known bounds and that $p(x, \mu_1)/p(x, \mu_0)$ be an increasing function of $x$ whenever $\mu_0 < \mu_1$. Complete classes are characterized for various loss functions $W(\mu, \alpha)$ which are convex functions of the decision $\alpha$ for each fixed value of $\mu$. Minimax procedures are considered for the case $W(\mu, \alpha) = |\alpha - \mu|^k$.