The Efficiency of Some Nonparametric Competitors of the t-Test
1956/06/01 by J. L. Hodges, E. L. Lehmann · 591 citations
Computer Science · Mathematics · #Asymptotic distribution #Bayesian Methods and Mixture Models #Combinatorics #Efficiency #Estimator #Function (biology) #Geometry #Limit (mathematics) #Mann–Whitney U test #Mathematical analysis #Mathematics #Nonparametric statistics #Power (physics) #Power function #Regular polygon #Sample size determination #Section (typography) #Sequence (biology) #Sign (mathematics) #Sign test #Statistical Methods and Inference #Statistical Methods in Clinical Trials #Statistics #Wilcoxon signed-rank test
paper · pdf · doi:10.1214/aoms/1177728261
published in The Annals of Mathematical Statistics 27(2), 324-335 (Institute of Mathematical Statistics)
openalex publication_date 1956/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/25
Abstract
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 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 → ∞ of the sequence of power efficiency functions. It is noted that certain interchanges of limit passages are not always possible.
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