Robustness of One-and Two-Sample Rank Tests Against Gross Errors

Abstract

The classical nonparametric hypotheses of symmetry and equality of distribution functions are extended to hypotheses of approximate symmetry and approximate equality, by allowing for gross errors, which is indispensable for practical applications. It is shown that for these hypotheses one- and two-sample rank statistics maintain their distribution freeness, which now refers to their stochastically extreme laws. These laws are evaluated asymptotically, also under similarly extended alternatives, in the author's previous local framework, which has not yet covered rank statistics due to a subtle asymptotic fine structure of infinitesimal neighborhoods. Consequences on asymptotic maximum size, minimum power and relative efficiency of rank tests are drawn. In particular, it is shown that if the scores are unbounded, then rank tests fail completely; and by suitable truncation of the classically optimal scores, an asymptotic maximin rank test is obtained.