Robust rank tests for k-sample approximate equality in the presence of gross errors

Abstract

Abstract The probability that a random vector of k-sample rank statistics takes the values in an arbitrary specified Schur-convex set in R k is considered under a k-sample local asymptotic gross error model. When the underlying distributions vary in gross error neighborhoods shrinking at the rate of n−1/2, lower and upper bounds for the limiting probability are obtained by using majorization methods on hyperplanes. These bounds enable us to construct asymptotic level α rank tests for k-sample approximate equality whose acceptance regions are Schur-convex sets, and to give lower bounds for their asymptotic minimum powers. For a class of k-sample rank tests an asymptotic relative efficiency is defined on the basis of the lower bounds and a k-sample robust rank test with a truncated scores generating function is proposed. It is shown that the asymptotic relative efficiency of the proposed rank test coincides with those in one- and two-sample approximate equality derived by Rieder (1981 , Ann. Statist. 9, 245–265). Two-sided robust rank tests for two-sample approximate equality are also considered.