Distributions of Some One-Sidedk-Sample Smirnov-Type Statistics

Abstract

Consider the problem of testing the equality of k populations in cases in which the populations can be ordered a priori from low to high on the basis of some underlying variable. This article proposes a new statistic based on the sums of the Smirnov-type statistic and provides evidence that the test based on this statistic has high power for a wide range of alternatives suggested by this problem. To construct the statistic, let F j (x) be the empirical distribution for the j th sample based on a random sample of size n (assumed to be the same for all populations), let D i + = sup x { F i (x) – F i +1 ( x )} and let S = Σ D i + . For k = 3, the exact null distribution of S is tabulated for n ≤ 30, and an asymptotic distribution is derived that gives adequate estimates of probability for n 's as small as 5. For k > 3, a procedure to derive the exact distribution of S is described that is based on summing probabilities associated with the joint distribution of D i + , i = 1, …, k − 1.