Distributions of Some One-Sidedk-Sample Smirnov-Type Statistics
Sylvan R. Wallenstein · Journal of the American Statistical Association · 1980
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 Fj (x) be the empirical distribution for the jth sample based on a random sample of size n (assumed to be the same for all populations), let Di + = sup x {Fi (x) – Fi +1(x)} and let S = Σ Di +. 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 Di +, i = 1, …, k − 1.