AN EXAMPLE OF A TWO-SIDED WILCOXON SIGNED RANK TEST WHICH IS NOT UNBIASED

PETER AMRHEIN

Seminar für Statistik, Universität Mannheim, A 5, 68131 Mannheim, Germany

(Received January 17, 1994; revised July 5, 1994)

Abstract. Although the theory of rank tests is rather complete in the onesided case, it was not even known in 1959, whether the Wilcoxon two-sample test and other similiar tests are unbiased against the two-sided alternatives (Lehmann (1959, Testing Statistical Hypotheses, p. 240, Wiley, New York)). A partial answer to this question was given by Sugiura in 1965, who found, that the test named above may be biased (Sugiura (1965, Ann. Inst. Statist. Math., 17, 261–263)). According to Lehmann (1986, Testing Statistical Hypotheses, 2nd ed., pp. 322–324, Wiley, New York) it seems to be still open, whether the same is true for the WILCOXON one-sample test, which is also known as WILCOXON signed rank test. This will be shown in the present paper.

Key words and phrases: Rank test, unbiasedness, counter-example.