On a Projection Estimator of the Regression Function Derivative

TL;DR

This paper investigates projection-based estimators for the derivative of a regression function, providing risk bounds, stability analysis, and an optimal rate in the compact case, along with a model selection procedure.

Contribution

It introduces a new projection estimator for the derivative of a regression function, with theoretical risk bounds and optimal rates, especially in the compact support case.

Findings

Risk bounds for the estimators are established.

Optimal rates are achieved in the compact case.

A model selection procedure with risk bounds is proposed.

Abstract

In this paper, we study the estimation of the derivative of a regression function in a standard univariate regression model. The estimators are defined either by derivating nonparametric least-squares estimators of the regression function or by estimating the projection of the derivative. We prove two simple risk bounds allowing to compare our estimators. More elaborate bounds under a stability assumption are then provided. Bases and spaces on which we can illustrate our assumptions and first results are both of compact or non compact type, and we discuss the rates reached by our estimators. They turn out to be optimal in the compact case. Lastly, we propose a model selection procedure and prove the associated risk bound. To consider bases with a non compact support makes the problem difficult.