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ASYMPTOTIC EXPANSIONS FOR TWO-STAGE RANK TESTS

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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 two-stage 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.

**Source**
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