On Probabilities of Excessive Deviations for Kolmogorov-Smirnov, Cramer-von Mises and Chi-Square Statistics

Abstract

Let $\alpha_n$ be the classical empirical process and $T: D\lbrack 0, 1\rbrack \rightarrow R$. Assume $T$ satisfies the Lipschitz condition. Using the Komlos-Major-Tusnady inequality, bounds for $P(T(\alpha_n) \geq x_n \sqrt n)$ are obtained for every $n$ and $x_n > 0$. Hence expansions for large deviations, as well as some moderate and Cramer-type large-deviations results for $T(\alpha_n)$, are derived.