SYMMETRIZED APPROXIMATE SCORE RANK TESTS FOR THE TWO-SAMPLE CASE

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

1Department of Statistics, Pennsylvania State University, 219 Pond Laboratory,

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

2Department 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.