Open Access
June, 1956 The Efficiency of Some Nonparametric Competitors of the $t$-Test
J. L. Hodges Jr, E. L. Lehmann
Ann. Math. Statist. 27(2): 324-335 (June, 1956). DOI: 10.1214/aoms/1177728261


Consider samples from continuous distributions $F(x)$ and $F(x - \theta)$. We may test the hypothesis $\theta = 0$ by using the two-sample Wilcoxon test. We show in Section 1 that its asymptotic Pitman efficiency, relative to the $t$-test, never falls below 0.864. This result also holds for the Kruskal-Wallis test compared with the $F$-test, and for testing the location parameter of a single symmetric distribution. A number of alternative notions of asymptotic efficiency are compared in Section 2. In this connection, certain difficulties arise because power is not necessarily a convex function of sample size. As an alternative to the Pitman notion of asymptotic efficiency, we consider in Section 3 one based on the speed with which power at a fixed alternative tends to 1. In particular we obtain, for the sign test relative to the $t$ in normal populations, the limit as $n \rightarrow \infty$ of the sequence of power efficiency functions. It is noted that certain interchanges of limit passages are not always possible.


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J. L. Hodges Jr. E. L. Lehmann. "The Efficiency of Some Nonparametric Competitors of the $t$-Test." Ann. Math. Statist. 27 (2) 324 - 335, June, 1956.


Published: June, 1956
First available in Project Euclid: 28 April 2007

zbMATH: 0075.29206
MathSciNet: MR79383
Digital Object Identifier: 10.1214/aoms/1177728261

Rights: Copyright © 1956 Institute of Mathematical Statistics

Vol.27 • No. 2 • June, 1956
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