A maximal inequality for local empirical processes under weak dependence

TL;DR

This paper develops a maximal inequality for local empirical processes under weak dependence, providing nonasymptotic bounds that improve understanding of estimation errors in dependent data scenarios.

Contribution

It introduces a new maximal inequality for local empirical processes under strong mixing conditions, accommodating complex function classes and applying to kernel density estimation.

Findings

Bounds control estimation error uniformly over function class, point, and bandwidth

Results apply to function classes with increasing complexity with sample size

Kernel density estimators under weak dependence achieve near-iid rates

Abstract

We introduce a maximal inequality for a local empirical process under strongly mixing data. Local empirical processes are defined as the (local) averages $\frac{1}{nh}\sum_{i=1}^n \mathbf{1}\{x - h \leq X_i \leq x+h\}f(Z_i)$, where $f$ belongs to a class of functions, $x \in \mathbb{R}$ and $h > 0$ is a bandwidth. Our nonasymptotic bounds control estimation error uniformly over the function class, evaluation point $x$ and bandwidth $h$. They are also general enough to accomodate function classes whose complexity increases with $n$. As an application, we apply our bounds to function classes that exhibit polynomial decay in their uniform covering numbers. When specialized to the problem of kernel density estimation, our bounds reveal that, under weak dependence with exponential decay, these estimators achieve the same (up to a logarithmic factor) sharp uniform-in-bandwidth rates derived in the iid setting by \cite{Einmahl2005}.