Improved Rates in the Empirical Bayes Monotone Multiple Decision Problem $\operatorname{MLR}$ Family

Abstract

In the context of the two-action, linear loss, exponential family multiple decision problem, Van Houwelingen (1973), (1976) has shown that faster rates of convergence are deducible for monotone empirical Bayes procedures than result from application of the bound established for general empirical Bayes procedures by Johns and Van Ryzin (1972). This note generalizes a (1973) Van Houwelingen bound to arbitrary $k$-action, monotone loss, MLR family multiple decision problems. An example is given to show that the result is a useful alternative to the recent Van Ryzin and Susarla (1977) multiple decision problem generalization of Johns and Van Ryzin.