Alternative Efficiencies for Signed Rank Tests

Abstract

Asymptotic efficiency curves for the one sample Wilcoxon and normal scores tests are obtained by comparing the exponential rate of convergence to zero of the type I error $(\alpha)$ while keeping the type II error $(\beta)$ fixed $(0 < \beta < 1)$. A wider than usual view of test performance consistent with small sample results is obtained. The Pitman efficiency value is derived as a special case when the alternative approaches the null hypothesis. Comparisons of the signed rank tests relative to $\bar{x}$ or $t$ for normal location alternatives yield small efficiency values for extreme alternatives. The relative performance of the Wilcoxon is seen to be slightly better than the normal scores for normal alternatives with larger location parameter values despite the local (Pitman) optimality of the normal scores. Similar results hold for two other non-normal alternatives considered.