The Distribution of the Maximum Deviation Between two Sample Cumulative Step Functions

Let x₁ < x₂ M < xₙ and y₁ < y₂ < < yₘ be the ordered results of two random samples from populations having continuous cumulative distribution functions F (x) and G (x) respectively. Let Sₙ (x) = K/n when k is the number of observed values of X which are less than or equal to x, and similarly let S'ₘ (y) = j/m where j is the number of observed values of Y which are less than or equal to y. The statistic d = | Sₙ (x) - S'ₘ (x) | can be used to test the hypothesis F (x) G (x), where the hypothesis would be rejected if the observed d is significantly large. The limiting distribution of d mnm + n has been derived 1 and 4, and tabled 5. In this paper a method of obtaining the exact distribution of d for small samples is described, and a short table for equal size samples is included. The general technique is that used by the author for the single sample case 2. There is a lower bound to the power of the test against any specified alternative, 3. This lower bound approaches one as n and m approach infinity proving that the test is consistent.