Uniform Estimation and Inference for Nonparametric Partitioning-Based M-Estimators

TL;DR

This paper develops comprehensive uniform estimation and inference methods for a broad class of nonparametric partitioning-based M-estimators, covering various models and improving upon existing theoretical results.

Contribution

It introduces new uniform consistency, Bahadur representations, and inference techniques for nonparametric M-estimators, extending to complex models and improving regularity conditions.

Findings

Achieves rate-optimal uniform convergence rates.

Provides valid strong approximation and inference methods.

Demonstrates improvements in quantile, distribution, and logistic regression examples.

Abstract

This paper presents uniform estimation and inference theory for a large class of nonparametric partitioning-based M-estimators. The main theoretical results include: (i) uniform consistency for convex and non-convex objective functions; (ii) rate-optimal uniform Bahadur representations; (iii) rate-optimal uniform (and mean square) convergence rates; (iv) valid strong approximations and feasible uniform inference methods; and (v) extensions to functional transformations of underlying estimators. Uniformity is established over both the evaluation point of the nonparametric functional parameter and a Euclidean parameter indexing the class of loss functions. The results also account explicitly for the smoothness degree of the loss function (if any), and allow for a possibly non-identity (inverse) link function. We illustrate the theoretical and methodological results in four examples: quantile regression, distribution regression, $L_p$ regression, and Logistic regression. Many other possibly non-smooth, nonlinear, generalized, robust M-estimation settings are covered by our results. We provide detailed comparisons with the existing literature and demonstrate substantive improvements: we achieve the best (in some cases optimal) known results under improved (in some cases minimal) requirements in terms of regularity conditions and side rate restrictions. The supplemental appendix reports complementary technical results that may be of independent interest, including a novel uniform strong approximation result based on Yurinskii's coupling.