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

The Cramer-von Mises ² criterion for testing that a sample, x₁, , xN, has been drawn from a specified continuous distribution F (x) is equation*1² = ^- FN (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 N², namely equation*2 T = NM/ (N + M) ^- FN (x) - GM (x) ² dH₍+₌ (x), equation* where GM (x) is the empirical distribution function of the second sample and H₍+₌ (x) is the empirical distribution function of the two samples together that is, (N + M) H₍+₌ (x) = NFN (x) + MGM (x). The limiting distribution of N² 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, where 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 N² 6). The accuracy of approximation is better than in the case of the two-sample Kolmogorov-Smirnov statistic studied by Hodges 4.