Asymptotic Comparison of Rank Tests for the Regression Problem When Ties are Present

Abstract

Without assumptions on the underlying distributions we prove the asymptotic normality of averaged scores rank statistics under all product distributions which are contiguous to the null hypothesis, and find a very simple form of the centering constants. In the one-sided two-sample and trend situations this enables us to show that monotonicity of the scores generating function is equivalent to the asymptotic unbiasedness of the corresponding averaged scores rank test. For such asymptotically unbiased tests we prove simple necessary and sufficient conditions for having bounds for their asymptotic relative efficiency under all contiguous alternatives of the model. As a by-product, we get the local asymptotic superiority of averaged scores rank tests to the associated tests with randomized ranks not only for shift but for general alternatives. In addition we prove that every one-sided averaged scores rank test is asymptotically most powerful (asymptotically equivalent to likelihood ratio test) for a suitable nonparametric subclass of alternatives, provided the test and the associated subclass of alternatives are generated by a nondecreasing, square-integrable function defined on the unit interval.