A Two-stage Rank Test Using Density Estimation

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-~, 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. (rv's) from a continuous distribution function (df) F(x-O) with F(-x) = 1-F(x) for all x. The hypothesis H0 : 0 = 0 is to be tested on the basis of a sample from the sequence X1, X~,.... In the classical case the problem is specialized to F(x) = ~(x/cr), where (I) is the standard normal dr. Stein's two-stage procedure for this situation is well-known (see e.g. Lehmann (1986), pp. 258-260). An initial sample of size rn is drawn and its sample variance S,2~ is obtained. In the second stage a sample of size N-m is selected, where (i.I) N = + 1), with c > 0 any given constant and [y] denoting the largest integer < y. Then N1/2(N-1 ~iNz Xi-O)/Sm has a tin_l-distribution and the constant c from (1.1) can be chosen such that the corresponding two-stage test has a prescribed power against a given alternative, regardless of the value of the scale parameter or. Now fixed sample size rank tests, being distributionfree, have levels independent of F, but powers directly dependent of F. Hence it seems worthwhile to try to 675 676 WILLEM ALBERS extend the property above of having powers independent (of certain aspects) of tile unknown underlying df, to the rank case. In fact, results in this direction have already been presented in Albers (1991, 1992). In these papers attention is focussed on a straightforward extension of Stein's procedure: using asymptotic expansions for local alternatives, a two-stage rank test is derived which has to o(m-1) a power independent of the scale parameter or. Of course, if we are not constrained to the classical case F(z) = ~(z/cr), there no longer is a special reason to let F …