The Theory of Decision Procedures for Distributions with Monotone Likelihood Ratio

Abstract

In many statistical decision problems, the observations can be summarized in a single sufficient statistic such that the likelihood ratio for any two distributions in the family under consideration is a monotone function of that statistic. This paper assumes, accordingly, that the statistician's decision is to be based upon a single observation of a random variable $X$, whose distribution is given by (1) and satisfies the inequality (2) in Section 1. As examples of this family of distributions, we have the exponential family such as the normal, binomial, and Poisson. Other kinds of examples are given in Section 1. In connection with the ordinary testing problem, Allen [1] showed that for the composite testing problem of the one-sided type for the special case of the exponential family of distributions, an admissible minimax procedure must be of the form: choose action 1 (accept the hypothesis) if $x < x_0$ and choose action 2 (accept the alternative) if $x > x_0$. If $x = x_0$, randomization may be required. Sobel [2] and Chernoff obtained partial results for the same class of distributions when the set of decisions is finite. This paper unifies, extends, and strengthens these results and treats of a wide variety of statistical decision problems for which the densities have a monotone likelihood ratio. In Section 1 the fundamental definition and preliminaries are introduced. In particular, the conditions imposed on the loss functions and the densities are delimited and some simple properties of these quantities are developed. In Section 2 we establish some of the basic lemmas. Noteworthy are Lemmas 1 and 2 which express the variation of sign diminishing properties of the densities which possess a monotone likelihood ratio. The essential completeness of the set of all monotone strategies (see Section 3 for the definition) in the class of all statistical procedures is demonstrated in Section 3 for the case of a finite number of actions. Section 4 deals with the problem of determining the form of all Bayes strategies for the statistician. The important problem of admissibility is studied in detail in Section 5. In the next section a study of the Bayes strategies for nature is made for the case of two actions. In Section 7 the complete class theory is carried through for the case of an infinite number of actions. This is accomplished by employing an argument involving a limiting procedure from the case of finite actions as treated in Section 3. The eighth section presents an analysis of the nature of the Bayes strategies for the case of an infinite number of actions. The final section entails a brief discussion of the connection of invariance theory and the conditions of monotonicity as are required throughout this paper. Further extensions of these ideas in a different direction, which involves relaxing the conditions on the loss functions and strengthening the requirements on the densities, can be found in [3].