Estimation in nonparametric regression model with additive and multiplicative noise via Laguerre series

TL;DR

This paper develops adaptive Laguerre series-based estimators for nonparametric regression with additive and multiplicative Gaussian noise, achieving near-optimal convergence rates under various noise structures.

Contribution

It introduces a novel Laguerre series approach for adaptive estimation in complex noise environments, extending existing methods to long-memory Gaussian errors.

Findings

Achieves near-optimal convergence rates for i.i.d. Gaussian noise.

Convergence rates depend on long-memory parameters under strong long-memory noise.

Rates are similar to classical results in the i.i.d. case.

Abstract

We look into the nonparametric regression estimation with additive and multiplicative noise and construct adaptive thresholding estimators based on Laguerre series. The proposed approach achieves asymptotically near-optimal convergence rates when the unknown function belongs to Laguerre-Sobolev space. We consider the problem under two noise structures; (1) { i.i.d.} Gaussian errors and (2) long-memory Gaussian errors. In the { i.i.d.} case, our convergence rates are similar to those found in the literature. In the long-memory case, the convergence rates depend on the long-memory parameters only when long-memory is strong enough in either noise source, otherwise, the rates are identical to those under { i.i.d.} noise.