The Large-Sample Power of Rank Order Tests in the Two-Sample Problem

Abstract

This paper studies the large-sample power of certain rank order tests against one-parameter alternatives in the two-sample problem. The first $m$ of $N$ independent random variables are supposed identically distributed, each with a density function $f_1(x, \theta)$, the remaining $N - m$ with a density function $f_2(x, \theta)$. When $\theta = 0$ both density functions are the same. Let $a_{N1}, \cdots, a_{NN}$ be a set of constants defined by (3.2) below; let $b_{N1}, \cdots, b_{NN}$ be another set of constants; and let $R_1, \cdots R_N$ be the ranks of the $N$ random variables. A statistic of the type $\sum^N_{i = 1} a_{Ni}b_{NR_i}$ is called an $L$ statistic. Part I of this paper characterizes the locally best rank order statistic for testing $H_0:\theta = 0$ against the alternative that $\theta$ is positive and "close" to zero. This turns out to be any one of an equivalent class of $L$ statistics. Under certain regularity conditions it is possible to determine the large-sample power of $L$ statistics. Of particular interest is the large-sample power of the locally best $L$ statistic. For arbitrary $b_{N1}, \cdots, b_{NN}$ it is usually difficult to determine whether the regularity conditions hold. Hence, in Part II a special class of $L$ statistics, the $L_h$ statistics, are studied. For these, the regularity conditions are easier to verify and the large-sample power is determined. The best $L$ statistic can, in a certain sense, be approximated by $L_h$ statistics.