A Bayesian Approach to Some Best Population Problems

Abstract

There have been several papers recently in the literature devoted to the subject of selecting a "best" population--see for example Gupta and Sobel (1962), Guttman (1961) and others. In these papers, the problem was analyzed from the sampling theory point of view. Except for some simple cases, this approach frequently leads to the problem of eliminating nuisance parameters. And, unless rather strong assumptions about certain of the parameters involved are made, the problem usually becomes intractable. In this paper, we consider certain best population problems adopting a Bayesian approach. As is well known, the modern Bayesian approach provides a logical framework for decision making under uncertainty--Savage (1954), Luce and Raiffa (1957), etc. Indeed, the adoption of this approach in the best population problem leads to intuitively satisfactory decision procedures in the presence of nuisance parameters. We consider a collection of $k$ populations $\prod_1, \cdots,\prod_i, \cdots, \prod_k$, where $\prod_i$ is distributed with probability density function $f(y \mid \theta_i)$ and $\theta_i$ may be vector-valued. Associated with the populations is a utility function $U(\theta_i)$. The best population is defined to be the one with the largest value among the $U(\theta_i)$. The Bayesian statistical decision procedure for choosing the best population is of course dictated by the principle of maximizing expected utility. Now it usually is the case that the experimenter's interest focuses on a specific criterion $h_i = g(\theta_i)$ where $g$ is known. For example, $g(\theta_i)$ might simply be the mean or the reciprocal of the variance of the $i$th population, $i = 1, \cdots, k$. It would then be natural for the experimenter to regard his utility function $U$ as a function of $h_i$, i.e., $U = U(h_i)$. In this paper, we will be mainly concerned with the case in which the criterion $g(\theta)$ is defined as $h = g(\theta) = \int^{a_2}_{a_1} f(y \mid \theta) dy$, for specified $a_1$ and $a_2$. The interval $(a_1, a_2)$ is sometimes referred to as a tolerance interval and the quantity $h$ is called the coverage of this interval. Considerations of tolerance intervals and their coverages frequently arise in engineering application [see for example Bowker and Lieberman (1959), and Ostle (1963)]. For instance, in the manufacture of a certain type of rivet, the product will only be of use if the diameter of the head of the rivet measures between certain specified limits. Or in the assembling of "stable" amplifiers, certain of the electronic tubes used in the amplifier must have transconductances that lie within specified limits. Hence, it is important for the manufacturer of these products to know what percentage of the items produced meet these required specifications. Suppose the manufacturer can choose among $k$ different processes to produce his product, and suppose further that his utility of the $i$th population $U(h_i) = h_i$ (that is, other considerations such as cost can be ignored). Naturally, he will wish to choose that process which gives the largest coverage to the specification interval. Of course, we are including the case that one process be capable of $k$ independent modifications. As an example, in the manufacture of precision items on a lathe, there may be a choice of different cutting tools, different coolants, and various speeds of rotation. Our interest in this type of problem is also motivated by the fact that the coverage criterion $h$ as defined above is usually a complicated function of the parameter(s) $\theta$. One might thus be led to suspect that it would in general be rather difficult to obtain optimal decision procedures for choosing the best population. In fact, one of the present authors--Guttman (1961)--has considered this problem within the sampling theory framework for the cases that $f$ takes the form of a normal density and that of an exponential density. It is evident from this earlier work that one would indeed encounter considerable mathematical difficulties unless certain specific assumptions about the scale parameters of the populations were made. We shall demonstrate, however, that no such difficulty exists in the Bayesian formulation and our analysis leads to results which seem intuitively satisfactory. In adopting a Bayesian approach, suppose we denote the prior distribution of the parameters of the $k$ populations under consideration as $p(\theta_1, \cdots, \theta_k)$. Then, for given independent samples from these populations, say $(\mathbf{y}_1, \cdots, \mathbf{y}_k)$, we can obtain the posterior distribution of these parameters. As we are interested in $(h_1, \cdots, h_k)$, which are themselves functions of the parameters, we may, in principle at least, determine their joint posterior distribution. This joint posterior distribution summarizes all the relevant information about $(h_1, \cdots, h_k)$. We assume throughout this paper that the population parameters $(\theta_1, \cdots, \theta_k)$ are locally independent a priori. This means that in the region in which the likelihood is appreciable, the joint distribution $p(\theta_1, \cdots, \theta_k)$ can be written approximately as the product $p_1(\theta_1) \cdots p_k(\theta_k)$. Such an assumption will be appropriate in situations where the prior distribution of the parameters is diffuse and gently changing over a wide region--see for example the discussion in Savage et al. (1962). It is clear that this independence assumption together with the assumption about the independence of samples implies that the $h$'s are locally independent a posteriori and hence can be analyzed individually. In Sections 2 and 3, we assume that the utility function $U(h_i)$ is $h_i$ for all $i$, and therefore analyze the problem by comparing the expected values of the posterior distribution of the coverages. The populations involved are assumed to be (1) normal and (2) exponential. The posterior distributions and the moments of the coverages for these populations are derived in Sections 4 and 5 where other types of utility functions are discussed. In Section 6 we briefly discuss some different types of best population problems.