The Annals of Mathematical Statistics

The Efficiency of Some Nonparametric Competitors of the $t$-Test

J. L. Hodges, Jr and E. L. Lehmann

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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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Ann. Math. Statist., Volume 27, Number 2 (1956), 324-335.

First available in Project Euclid: 28 April 2007

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Hodges, J. L.; Lehmann, E. L. The Efficiency of Some Nonparametric Competitors of the $t$-Test. Ann. Math. Statist. 27 (1956), no. 2, 324--335. doi:10.1214/aoms/1177728261.

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