MODERATELY LARGE DEVIATIONS AND EXPANSIONS OF LARGE DEVIATIONS FOR SOME FUNCTIONALS OF WEIGHTED EMPIRICAL PROCESSES

Let alpha(n) be the classical empirical process. Assume T, defined on D[0, 1], satisfies the Lipschitz condition with respect to a weighted sup-norm in D[0, 1]. Explicit bounds for P(T(alpha(n)) greater-than-or -equal-to x(n) square-root n) are obtained for every n greater-than-or -equal-to n0 and all x(n) is-an-element-of (0, sigma], where n0 and si gma are also explicitly given. These bounds lead to moderately large d eviations and expansions of the asymptotic large deviations for T(alph a(n)). The present theory closely relates large and moderately large d eviations to tails of the asymptotic distributions of considered stati stics. It unifies and generalizes some earlier results. In particular, some results of Groeneboom and Shorack are easily derived.