Variance estimation in nonparametric regression via the difference sequence method

TL;DR

This paper introduces difference-based kernel estimators for variance functions in Gaussian nonparametric regression, establishing their optimal convergence rates, asymptotic normality, and minimax optimality across broad function classes.

Contribution

It proposes a new class of difference-based kernel estimators for variance functions and characterizes their optimal convergence and asymptotic properties.

Findings

Estimators achieve optimal convergence rates.

Asymptotic normality is established.

Estimators attain the minimax rate.

Abstract

Consider a Gaussian nonparametric regression problem having both an unknown mean function and unknown variance function. This article presents a class of difference-based kernel estimators for the variance function. Optimal convergence rates that are uniform over broad functional classes and bandwidths are fully characterized, and asymptotic normality is also established. We also show that for suitable asymptotic formulations our estimators achieve the minimax rate.