On the Theory of Some Non-Parametric Hypotheses

Abstract

For two types of non-parametric hypotheses optimum tests are derived against certain classes of alternatives. The two kinds of hypotheses are related and may be illustrated by the following example: (1) The joint distribution of the variables $X_1, \cdots, X_m, Y_1, \cdots, Y_n$ is invariant under all permutations of the variables; (2) the variables are independently and identically distributed. It is shown that the theory of optimum tests for hypotheses of the first kind is the same as that of optimum similar tests for hypotheses of the second kind. Most powerful tests are obtained against arbitrary simple alternatives, and in a number of important cases most stringent tests are derived against certain composite alternatives. For the example (1), if the distributions are restricted to probability densities, Pitman's test based on $\bar y - \bar x$ is most powerful against the alternatives that the $X$'s and $Y$'s are independently normally distributed with common variance, and that $E(X_i) = \xi, E(Y_i) = \eta$ where $\eta > \xi$. If $\eta - \xi$ may be positive or negative the test based on $|\bar y - \bar x|$ is most stringent. The definitions are sufficiently general that the theory applies to both continuous and discrete problems, and that tied observations present no difficulties. It is shown that continuous and discrete problems may be combined. Pitman's test for example, when applied to certain discrete problems, coincides with Fisher's exact test, and when $m = n$ the test based on $|\bar y - \bar x|$ is most stringent for hypothesis (1) against a broad class of alternatives which includes both discrete and absolutely continuous distributions.