A Stochastic Ordering Induced by a Concept of Positive Dependence and Monotonicity of Asymptotic Test Sizes

Abstract

An ordering of distributions related to a concept of positive dependence is studied and stochastic monotonicity with respect to this ordering is established for a wide class of two-sample test statistics. Asymptotic conservativeness of test sizes under certain departures from independence between samples is discussed. For example, if the observations are paired and the joint density is positive semidefinite then tests such as Kolmogorov-Smirnov, $\chi^2$ and Cramer-von Mises, as well as a large class of linear rank tests, are shown to be asymptotically conservative.