A local maximal inequality under uniform entropy

TL;DR

This paper establishes a new upper bound for the supremum of empirical processes based on uniform entropy, aiding in understanding the convergence rates of estimators in statistical learning.

Contribution

It introduces a novel maximal inequality under uniform entropy conditions, providing sharper bounds for empirical processes with bounded variance.

Findings

Derived an upper bound for empirical process supremum using uniform entropy

Provided convergence rate results for minimum contrast estimators

Enhanced understanding of empirical process behavior under variance constraints

Abstract

We derive an upper bound for the mean of the supremum of the empirical process indexed by a class of functions that are known to have variance bounded by a small constant $\delta$. The bound is expressed in the uniform entropy integral of the class at $\delta$. The bound yields a rate of convergence of minimum contrast estimators when applied to the modulus of continuity of the contrast functions.