The Efficiency of Some Nonparametric Competitors of the t-Test

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.