Starshaped Transformations and the Power of Rank Tests

Abstract

It is known (Lehmann (1959), p. 187, and Bell, Moser and Thompson (1966), p. 134) that for the two-sample problem, the closer the two samples are together stochastically, the smaller is the power of monotone rank tests. Here it is shown that if one uses the ideas of van Zwet (1964) to define "skewness" and "heavy tails," then the more skew the distributions of the two samples are, the smaller is the power of monotone rank tests; and heavy tails similarly leads to smaller power of monotone rank tests. Skewness and heavy tails are defined using convex and star-shaped transformations of random variables. These are the same transformations used in reliability theory (Barlow and Proschan (1965), Birnbaum, Esary and Marshall (1966), and others) to describe the concept of "wear-out." Thus if $X$ is a random variable that represents "time to failure," and if failure is caused by wear-out or by the environment, then there exists a convex or starshaped function $g$ such that $Z = g(X)$ is an exponential $(1 - \exp \lbrack -\lambda z \rbrack)$ random variable. The distributions of these variables $X$ are called increasing failure rate (IFR) distributions when $g$ is convex and (IFRA) distributions when $g$ is starshaped. It turns out that if one restricts attention to such distributions, then the results of this paper can be used to construct a simple optimality theory for rank tests. This is done in a later paper [6]. The power inequalities related to skewness and heavy tails readily extends to sequential rank tests. It is shown (Example 5.1) that the sequential probability ratio test based on ranks for exponential scale alternatives (e.g. [11] and [12]) also is valid for the class of IFRA scale alternatives.