Monotonicity of Rank Statistics in Some Nonparametric Testing Problems

摘要

For nonparametric testing problems (one-sided two-sample problem, k-sample trend problem, testing independence against positive dependence) a partial ordering over the alternatives is defined. This partial ordering expresses the strength of the deviation from the null-hypothesis. All familiar rank tests turn out to be more powerful under increasing alternatives; that is, all familiar rank statistics preserve the ordering stochastically in samples whenever it is present between underlying distributions. As a tool, the sample equivalence of the ordering is introduced as a partial ordering over pairs of permutations. Functions, defined on pairs of permutations, which preserve this ordering are studied.