On a Problem in the Theory of k Populations

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 π₁, π₂, ⋯, πₖ be a set of normal populations, πᵢ having an unknown mean mᵢ and variance σ², G(x, θᵢ) being the distribution function which characterizes πᵢ. Samples of equal size are drawn from each population, Xᵢ being the sample means, and S² the estimate of σ² obtained. The problem is to construct a suitable decision rule d = d(\ Xᵢ\; S²) to select one or more populations, the object being to minimize the expected value of the random distribution function G(x s(d)) = ∑ᵏᵢ₌₁ Zᵢ(d) · G(x, θᵢ) / ∑ᵏᵢ₌₁ Zᵢ(d), where Zᵢ(d) = 1 if πᵢ is selected by d, and $= 0$ otherwise. It is shown that under the restriction of impartial decision, the rule dₖ = "Always select only the population corresponding to the greatest Xᵢ" cannot be improved, no matter what x or the true parameter values may be. It follows (i) that dₖ is the uniformly best decision rule in the class of impartial decision rules for all weight functions of type W = maxᵢ ᵢ\ - (∑ᵏᵢ₌₁ zᵢmᵢ / ∑ᵏᵢ₌₁ zᵢ), 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.