ASYMPTOTIC EXPANSIONS FOR TWO-STAGE RANK TESTS

WILLEM ALBERS

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

(Received June 28, 1990; revised March 11, 1991)

Abstract. Stein’s two-stage procedure produces a t-test which can realize a prescribed power against a given alternative, regardless of the unknown variance of the underlying normal distribution. This is achieved by determining the size of a second sample on the basis of a variance estimate derived from the first sample. In the paper we introduce a nonparametric competitor of this classical procedure by replacing the t-test by a rank test. For rank tests, the most precise information available are asymptotic expansions for their power to order n^{−1}, where n is the sample size. Using results on combinations of rank tests for sub-samples, we obtain the same level of precision for the twostage case. In this way we can determine the size of the additional sample to the natural order and moreover compare the nonparametric and the classical procedure in terms of expected additional numbers of observations required.

Key words and phrases: One-sample problem, Stein’s two-stage procedure.