On the Limiting Distribution of and Critical Values for the Multivariate Cramer-Von Mises Statistic

Abstract

Let $Y_1, Y_2, \cdots, Y_n (n = 1, 2, \cdots)$ be independent random variables (r.v.'s) uniformly distributed over the $d$-dimensional unit cube, and let $\alpha_n(\cdot)$ be the empirical process based on this sequence of random samples. Let $V_{n, d}(\cdot)$ be the distribution function of the Cramer-von Mises functional of $\alpha_n(\cdot)$, and define $V_d(\cdot) = \lim_{n \rightarrow \infty} V_{n, d}(\cdot), \Delta_{n, d} = \sup_{0 < x < \infty}|V_{n, d}(x) - V_d(x)|$. We deduce that $\Delta_{n,d} = O(n^{-1}), d \geq 1$, and calculate also the "usual" levels of significance of the distribution function $V_d(\cdot)$ for $d = 2$ to 50, using expansion methods. Previously these were known only for $d = 1, 2, 3$.