Strong Approximations for Empirical Processes Indexed by Lipschitz Functions

TL;DR

This paper develops advanced Gaussian strong approximation techniques for empirical processes indexed by Lipschitz functions, improving approximation rates and extending results to multivariate settings, with applications in nonparametric estimation.

Contribution

It introduces new uniform Gaussian strong approximations for empirical processes indexed by Lipschitz functions, improving rates and extending to multivariate and multiplicative cases.

Findings

Improved approximation rate from $n^{-1/(2d)}$ to $n^{-1/ ext{max}\{d,2 ight ext{}}$ for Lipschitz functions.

Established a $n^{-1/2} ext{log} n$ rate for $d=2$, previously known only for $d=1$.

Provided new results for empirical processes indexed by Haar basis functions.

Abstract

This paper presents new uniform Gaussian strong approximations for empirical processes indexed by classes of functions based on $d$-variate random vectors ($d\geq1$). First, a uniform Gaussian strong approximation is established for general empirical processes indexed by possibly Lipschitz functions, improving on previous results in the literature. In the setting considered by Rio (1994), and if the function class is Lipschitzian, our result improves the approximation rate $n^{-1/(2d)}$ to $n^{-1/\max\{d,2\}}$, up to a $\operatorname{polylog}(n)$ term, where $n$ denotes the sample size. Remarkably, we establish a valid uniform Gaussian strong approximation at the rate $n^{-1/2}\log n$ for $d=2$, which was previously known to be valid only for univariate ($d=1$) empirical processes via the celebrated Hungarian construction (Koml\'os et al., 1975). Second, a uniform Gaussian strong approximation is established for multiplicative separable empirical processes indexed by possibly Lipschitz functions, which addresses some outstanding problems in the literature (Chernozhukov et al., 2014, Section 3). Finally, two other uniform Gaussian strong approximation results are presented when the function class is a sequence of Haar basis based on quasi-uniform partitions. Applications to nonparametric density and regression estimation are discussed.