A TWO-STAGE RANK TEST USING DENSITY ESTIMATION

WILLEM ALBERS

Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

(Received February 28, 1994; revised March 31, 1995)

Abstract. For the one-sample problem, a two-stage rank test is derived which realizes a required power against a given local alternative, for all sufficiently smooth underlying distributions. This is achieved using asymptotic expansions resulting in a precision of order m^{−1}, where m is the size of the first sample. The size of the second sample is derived through a number of estimators of e.g. integrated squared densities and density derivatives, all based on the first sample. The resulting procedure can be viewed as a nonparametric analogue of the classical Stein’s two-stage procedure, which uses a t-test and assumes normality for the underlying distribution. The present approach also generalizes earlier work which extended the classical method to parametric families of distributions.

Key words and phrases: One-sample tests, local alternatives, Stein’s two-stage procedure, Wilcoxon scores.