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SYMMETRIZED APPROXIMATE SCORE RANK TESTS

FOR THE TWO-SAMPLE CASE

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MICHAEL G. AKRITAS^{1} AND RICHARD A. JOHNSON^{2}

^{1} *Department of Statistics, Pennsylvania State University, 219 Pond Laboratory,*

University Park, PA 16802, U.S.A.

^{2} *Department of Statistics, University of Wisconsin, 1210 West Dayton Street,*

Madison, WI 53706, U.S.A.
(Received September 11, 1990; revised November 11, 1991)

**Abstract.**
Rank test statistics for the two-sample problem are
based on
the sum of the rank scores from either sample. However, a critical
difference can
occur when approximate scores are used since the sum of the rank scores
from sample 1
is not equal to minus the sum of the rank scores from sample 2. By centering and
scaling as described in Hajek and Sidak (1967, *Theory of Rank Tests*,
Academic Press, New York) for the uncensored data case the statistics
computed from
each sample become identical. However such symmetrized approximate scores rank
statistics have not been proposed in the censored data case. We propose a
statistic
that treats the two approximate scores rank statistics in a symmetric
manner. Under
equal censoring distributions the symmetric rank tests are efficient when
the score
function corresponds to the underlying model distribution. For unequal censoring
distributions we derive a useable expression for the asymptotic variance of our
symmetric rank statistics.

*Key words and phrases*:
Two-sample problem, approximate
scores, Pitman
efficiency, unequal censoring, Skorokhod construction.

**Source**
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