Effect of mean on variance function estimation in nonparametric regression

TL;DR

This paper investigates how the unknown mean function affects variance estimation in nonparametric regression, revealing that using minimal bias mean estimators is preferable over residual-based methods when the mean is not smooth.

Contribution

It provides the first minimax rate of convergence for variance function estimation considering the mean's effect and corrects previous rate claims.

Findings

Residual-based estimators perform poorly with non-smooth means.

Minimal bias mean estimators improve variance estimation accuracy.

Corrects the optimal rate previously claimed by Hall and Carroll.

Abstract

Variance function estimation in nonparametric regression is considered and the minimax rate of convergence is derived. We are particularly interested in the effect of the unknown mean on the estimation of the variance function. Our results indicate that, contrary to the common practice, it is not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean when the mean function is not smooth. Instead it is more desirable to use estimators of the mean with minimal bias. On the other hand, when the mean function is very smooth, our numerical results show that the residual-based method performs better, but not substantial better than the first-order-difference-based estimator. In addition our asymptotic results also correct the optimal rate claimed in Hall and Carroll [J. Roy. Statist. Soc. Ser. B 51 (1989) 3--14].