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

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.

Keywords:

• Nonparametric regression
• Laguerre series
• Laguerre–Sobolev space
• long-memory
• minimax convergence rate

AMS (2000) Subject Classification:

• 62G05
• 62G20
• 62G08

Acknowledgments

The author would like to thank the Editor and an anonymous referee for their useful comments and suggestions which have led to the improved version of the paper.