Spatial adaptation in heteroscedastic regression: Propagation approach

TL;DR

This paper develops and justifies a propagation-based adaptive estimation method for heteroscedastic regression with noise misspecification, achieving robustness with a relative error of order 1/log(n).

Contribution

It extends the propagation approach to heteroscedastic regression, providing a theoretically justified method for adaptive estimation under noise misspecification.

Findings

Adaptive procedure tolerates covariance matrix misspecification of order 1/log(n).

Method is based on local approximation and Lepski's method.

The approach is theoretically justified for heteroscedastic noise.

Abstract

The paper concerns the problem of pointwise adaptive estimation in regression when the noise is heteroscedastic and incorrectly known. The use of the local approximation method, which includes the local polynomial smoothing as a particular case, leads to a finite family of estimators corresponding to different degrees of smoothing. Data-driven choice of localization degree in this case can be understood as the problem of selection from this family. This task can be performed by a suggested in Katkovnik and Spokoiny (2008) FLL technique based on Lepski's method. An important issue with this type of procedures - the choice of certain tuning parameters - was addressed in Spokoiny and Vial (2009). The authors called their approach to the parameter calibration "propagation". In the present paper the propagation approach is developed and justified for the heteroscedastic case in presence of the noise misspecification. Our analysis shows that the adaptive procedure allows a misspecification of the covariance matrix with a relative error of order 1/log(n), where n is the sample size.