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

• J. L. HodgesJr.
• E. L. Lehmann
Open Access
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

## Summary

Consider samples from continuous distributions F(x) and F(x — θ). We may test the hypothesis θ = 0 by using the two-sample Wilcoxon test. We show in Section 1 that its asymptotic Pitman efficiency, relative to the f-test, never falls below 0.864. This result also holds for the Kruskal-Wallis test compared with the jF-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 —► ∞ of the sequence of power efficiency functions. It is noted that certain interchanges of limit passages are not always possible.

## Keywords

Sign Test Normal Test Asymptotic Efficiency Consecutive Integer Limit Passage
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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## Authors and Affiliations

• J. L. HodgesJr.
• 1
• E. L. Lehmann
• 1
1. 1.University of CaliforniaBerkeleyUSA