Rank Tests for One Sample, Two Samples, and $k$ samples Without the Assumption of a Continuous Distribution Function

Abstract

The theory of rank tests has been developed primarily for continuous random variables. Recently the asymptotic theory of linear rank tests has been extended to include purely discrete random variables under the null hypothesis of randomness (including the two-sample and $k$-sample problems) and under contiguous alternatives, for the two methods of assigning scores known as the average scores method and the randomized ranks method. In this paper the theory of rank tests is developed with no assumptions concerning the continuous or discrete nature of the underlying distribution function. Conditional rank tests, given the vector of ties, are shown to be similar, and the locally most powerful conditional rank test is given. The asymptotic distribution of linear rank statistics is given under the null hypotheses of randomness and symmetry (which includes the one-sample problem), and under contiguous alternatives. Three methods of assigning scores, the average scores, midranks, and randomized ranks methods, are discussed and briefly compared.