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

Abstract

The Cramer-von Mises $\omega^2$ criterion for testing that a sample, $x_1, \cdots, x_N$, has been drawn from a specified continuous distribution $F(x)$ is \begin{equation*}\tag{1}\omega^2 = \int^\infty_{-\infty} \lbrack F_N(x) - F(x)\rbrack^2 dF(x),\end{equation*} where $F_N(x)$ is the empirical distribution function of the sample; that is, $F_N(x) = k/N$ if exactly $k$ observations are less than or equal to $x(k = 0, 1, \cdots, N)$. If there is a second sample, $y_1, \cdots, y_M$, a test of the hypothesis that the two samples come from the same (unspecified) continuous distribution can be based on the analogue of $N\omega^2$, namely \begin{equation*}\tag{2} T = \lbrack NM/(N + M)\rbrack \int^\infty_{-\infty} \lbrack F_N(x) - G_M(x)\rbrack^2 dH_{N+M}(x),\end{equation*} where $G_M(x)$ is the empirical distribution function of the second sample and $H_{N+M}(x)$ is the empirical distribution function of the two samples together [that is, $(N + M)H_{N+M}(x) = NF_N(x) + MG_M(x)\rbrack$. The limiting distribution of $N\omega^2$ as $N \rightarrow \infty$ has been tabulated [2], and it has been shown ([3], [4a], and [7]) that $T$ has the same limiting distribution as $N \rightarrow \infty, M \rightarrow \infty$, and $N/M \rightarrow \lambda$, where $\lambda$ 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 $N\omega^2$ [6]). The accuracy of approximation is better than in the case of the two-sample Kolmogorov-Smirnov statistic studied by Hodges [4].