An Elementary Method for Obtaining Lower Bounds on the Asymptotic Power of Rank Tests

Abstract

The rapid development of non-parametric rank tests was generated, in part, by the result of Hodges and Lehmann [6] which stated that the asymptotic relative efficiency (ARE) of the Wilcoxon test to the classical two-sample $t$-test was always $\geqq.86$. They also conjectured that the ARE of the normal scores test to the $t$-test was always greater than or equal to one. In 1958, Chernoff and Savage [3] proved the validity of this conjecture using variational methods. In this paper we give a simple proof of their result. Recently Doksum [4] has shown that the Savage test [11] maximizes the minimum asymptotic power for testing for scale change over the family of distributions with increasing failure rate averages (IFRA) [2]. The technique of our proof enables us to obtain a lower bound for the asymptotic power of Savage's test for scale change of any positive random variable possessing a finite second moment. When the positive random variables are restricted to be IFRA, Doksum's [4] results follow.