Adaptive asymptotically efficient estimation in heteroscedastic nonparametric regression

TL;DR

This paper proves that an adaptive estimation procedure for heteroscedastic nonparametric regression is asymptotically efficient, achieving the optimal quadratic risk as defined by the Pinsker constant.

Contribution

It establishes the asymptotic efficiency of a previously proposed adaptive estimator in heteroscedastic nonparametric regression models.

Findings

The estimator attains the Pinsker constant asymptotically.

The procedure is proven to be asymptotically optimal for quadratic risk.

The results extend the understanding of adaptive estimation in heteroscedastic settings.

Abstract

The paper deals with asymptotic properties of the adaptive procedure proposed in the author paper, 2007, for estimating an unknown nonparametric regression. %\cite{GaPe1}. We prove that this procedure is asymptotically efficient for a quadratic risk, i.e. the asymptotic quadratic risk for this procedure coincides with the Pinsker constant which gives a sharp lower bound for the quadratic risk over all possible estimates