A Characterization of Certain Rank-Order Tests with Bounds for the Asymptotic Relative Efficiency

Abstract

For the one-sample independence problem, the one-sample symmetry problem, and the two-sample problem it is shown that every one-sided rank test is asymptotically optimal for a certain nonparametric subclass of contiguous alternatives, provided the test and the associated subclass of alternatives are generated by certain square-integrable functions defined on the unit square. Then the asymptotic normality of the respective rank statistics under every alternative contiguous to the hypothesis is used in order to give necessary and sufficient conditions for local asymptotic unbiasedness of such tests. Finally, for locally asymptotically unbiased tests there are given necessary and sufficient conditions for having bounds for their asymptotic relative efficiency under contiguous alternatives.