On a Problem in the Theory of k Populations

Abstract

In two recent papers, Paulson [1] and Mosteller [2] have called attention to several unsolved problems in $k$-sample theory. A problem which is typical of the ones considered in this paper is as follows. Let $\pi_1, \pi_2, \cdots, \pi_k$ be a set of normal populations, $\pi_i$ having an unknown mean $m_i$ and variance $\sigma^2, G(x, \theta_i)$ being the distribution function which characterizes $\pi_i$. Samples of equal size are drawn from each population, $\bar X_i$ being the sample means, and $S^2$ the estimate of $\sigma^2$ obtained. The problem is to construct a suitable decision rule $d = d(\{\bar X_i\}; S^2)$ to select one or more populations, the object being to minimize the expected value of the random distribution function $G(x \mid s(d)) = \sum^k_{i=1} Z_i(d) \cdot G(x, \theta_i) / \sum^k_{i=1} Z_i(d),$ where $Z_i(d) = 1$ if $\pi_i$ is selected by $d$, and $= 0$ otherwise. It is shown that under the restriction of impartial decision, the rule $d_k =$ "Always select only the population corresponding to the greatest $\bar X_i$" cannot be improved, no matter what $x$ or the true parameter values may be. It follows (i) that $d_k$ is the uniformly best decision rule in the class of impartial decision rules for all weight functions of type $W = \max_i \{m_i\} - \big(\sum^k_{i=1} z_im_i / \sum^k_{i=1} z_i\big)$, and (ii) that the customary $F$ and $t$ tests of analysis of variance are not relevant to the problem. This result is an application of Theorem 1 which applies to a number of similar problems concerning $k$ populations, especially when the populations admit sufficient statistics for their parameters. Two examples of statistical applications are given in Section 6.