Some Distribution-Free Tests for the Difference Between two Empirical Cumulative Distribution Functions

Abstract

It sometimes happens that of two empirical cumulative distribution curves (step curves) one lies entirely above the other, in other words that, except at both ends, they have no point in common. The problem then arises, what is the probability that this will happen when both are random samples from the same population. In this paper a partial answer will be given, based on the ingenious solution of Andre (as cited in the well known textbook of Bertrand [1] in the problem of the ballot and also in Chap. VIII, Sect. 5 of [7]). Moreover an analogous method will allow us to give an exact answer to the problem of the maximum difference between two empirical cumulative distribution functions of random samples from the same population, but only if both samples have the same size. Smirnov has given an asymptotic solution for the latter problem (cited by Feller [3], see also [2]). Our result leads, by using the Stirling approximation for the factorials, to the asymptotic formula of Smirnov. A comparison of numerical results of the exact formula and the asymptotic formula of Smirnov shows that at least in the case of equal samples, the probabilities calculated by the Smirnov formula have, for samples as small as 20, an error of less than 4% for probabilities 0.033 or more. (See also Massey [5], who has calculated the exact probabilities for equal samples by means of difference equations.)