Asymptotically Minimax Distribution-free Procedures

Abstract

The power and efficiency of robust procedures have been considered for parametric alternatives by Tukey (1946), Hoeffding (1951), Lehmann (1953), (1959), Chernoff and Savage (1958), Capon (1961), and others. For instance, it has been shown that the two-sample normal scores statistic is optimal for normal translation alternatives in that for these alternatives, it is locally most powerful [12], [15], [4] in the class of all rank statistics and it is asymptotically efficient [6], [4] in the class of all statistics. However, for other types of translation alternatives, the normal scores statistic does not have these optimal properties. In this paper, optimality properties for non-parametric classes of alternatives are treated. In particular, statistics that maximize the minimum power asymptotically over classes of non-parametric alternatives will be considered. If one takes (1-power) as the risk function, these statistics are asymptotically minimax. It turns out that in this sense the Wilcoxon statistic is asymptotically minimax over a class of one-sided alternatives $(F, G)$ with $\rho(F, G) \geqq \Delta$, where $\rho$ is the Kolmogorov distance, while the normal scores statistic is asymptotically minimax for a non-parametric class of translation type alternatives. For the problem of estimating differences in location, the normal scores estimate of Hodges and Lehmann (1963) is shown to be minimax when the risk function is asymptotic variance. Finally, the results are used to obtain asymptotic efficiencies that are defined for non-parametric classes of alternatives.