Moderately Large Deviations and Expansions of Large Deviations for Some Functionals of Weighted Empirical Processes

Abstract

Let $\alpha_n$ be the classical empirical process. Assume $T$, defined on $D\lbrack 0, 1\rbrack$, satisfies the Lipschitz condition with respect to a weighted sup-norm in $D\lbrack 0, 1\rbrack$. Explicit bounds for $P(T(\alpha_n) \geq x_n\sqrt n)$ are obtained for every $n \geq n_0$ and all $x_n \in (0, \sigma\rbrack$, where $n_0$ and $\sigma$ are also explicitly given. These bounds lead to moderately large deviations and expansions of the asymptotic large deviations for $T(\alpha_n)$. The present theory closely relates large and moderately large deviations to tails of the asymptotic distributions of considered statistics. It unifies and generalizes some earlier results. In particular, some results of Groeneboom and Shorack are easily derived.