On the Distribution of the Two-Sample Cramer-von Mises Criterion

The Cramer-von Mises ω² criterion for testing that a sample, x₁, ⋯, xN, has been drawn from a specified continuous distribution $F(x)$ is {equation*}{1}ω^2 = ∫^∞-∞ F_N(x) - F(x)^2 dF(x),{equation*} where FN(x) is the empirical distribution function of the sample; that is, FN(x) = k/N if exactly k observations are less than or equal to x(k = 0, 1, ⋯, N). If there is a second sample, y₁, ⋯, yM, a test of the hypothesis that the two samples come from the same (unspecified) continuous distribution can be based on the analogue of Nω², namely {equation*}{2} T = NM/(N + M) ∫^∞-∞ F_N(x) - G_M(x)^2 dHN+M(x),{equation*} where GM(x) is the empirical distribution function of the second sample and HN+M(x) is the empirical distribution function of the two samples together [that is, (N + M)HN+M(x) = NFN(x) + MGM(x). The limiting distribution of Nω² as N → ∞ has been tabulated [2], and it has been shown ([3], [4a], and [7]) that T has the same limiting distribution as N → ∞, M → ∞, and N/M → λ, where λ is any finite positive constant. In this note we consider the distribution of T for small values of N and M and present tables to permit use of the criterion at some conventional significance levels for small values of N and M. The limiting distribution seems a surprisingly good approximation to the exact distribution for moderate sample sizes (corresponding to the same feature for Nω² [6]). The accuracy of approximation is better than in the case of the two-sample Kolmogorov-Smirnov statistic studied by Hodges [4].