Bounds for Weighted Empirical Distribution Functions

Abstract

Let $G_n$ be the empirical distribution based on $n$ independent uniform random variables. Criteria for bounds on the supremum of weighted discrepancies between $G_n(u)$ and $u$ of the form: $|w_\nu(u) D_n(u)|$, where $D_n(u) = G_n(u) - u, w_\nu(u) = (u(1 - u))^{-1 + \nu}$ and $0 \leq \nu \leq 1$, are derived. Also an inequality closely related to an equality due to Daniels (1945) is given.