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

Abstract

Let $x_1 < x_2 M \cdots < x_n$ and $y_1 < y_2 < \cdot < y_m$ be the ordered results of two random samples from populations having continuous cumulative distribution functions $F(x)$ and $G(x)$ respectively. Let $S_n(x) = K/n$ 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 $d = \max | S_n(x) - S'_m(x) |$ can be used to test the hypothesis $F(x) \equiv G(x)$, where the hypothesis would be rejected if the observed $d$ is significantly large. The limiting distribution of $d \sqrt{mn}{m + n}$ 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.