Bounds in $L^1$ Wasserstein distance on the normal approximation of general M-estimators

TL;DR

This paper establishes near-optimal bounds on the convergence rate of general M-estimators in $L^1$ Wasserstein distance, even with dependent data, and applies these results to covariance parameter estimation in Gaussian processes.

Contribution

It provides a novel quantitative analysis of the convergence rates of M-estimators without explicit formulas, extending to dependent observations and applying to Gaussian process covariance estimation.

Findings

Derived near-sharp convergence bounds in $L^1$ Wasserstein distance.

Applicable to dependent data and estimators without explicit formulas.

Demonstrated effectiveness in Gaussian process covariance parameter estimation.

Abstract

We derive quantitative bounds on the rate of convergence in $L^1$ Wasserstein distance of general M-estimators, with an almost sharp (up to a logarithmic term) behavior in the number of observations. We focus on situations where the estimator does not have an explicit expression as a function of the data. The general method may be applied even in situations where the observations are not independent. Our main application is a rate of convergence for cross validation estimation of covariance parameters of Gaussian processes.