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 = max | 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.
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Frank J. Massey (1951) studied this question.