A covariate-matched estimator of the error variance in nonparametric regression

Abstract

There are two classes of estimators for the error variance in nonparametric regression: residual-based estimators and difference-based estimators. Residual-based estimators require an estimator of the regression function and are asymptotically equivalent to the sample variance based on the actual errors. Difference-based estimators avoid estimating the regression function and are thus simpler to calculate. They also possess superior bias properties at the expense of larger variances. Müller et al. [U.U. Müller, A. Schick, and W. Wefelmeyer, Estimating the error variance in nonparametric regression by a covariate-matched U-statistics, Statistics 37 (2003), pp. 179–188.] suggested improving difference-based estimators using covariate matching. They showed that a covariate-matched version of Rice's [J. Rice, Bandwidth choice for nonparametric regression, Ann. Statist. 12 (1984), pp. 1215–1230.] difference-based estimator matches the asymptotic performance of residual-based estimators, yet still possesses the good bias properties of Rice's estimator. Here we prove a similar result for a covariate-matched version of the difference-based estimator of Gasser et al. [T. Gasser, L. Sroka, and C. Jennen-Steinmetz, Residual variance and residual pattern in nonlinear regression, Biometrika 73 (1986), pp. 625–633.] as their estimator has even better bias properties than Rice's estimator.

Keywords:

• difference-based estimator
• U-process
• asymptotic normality

Acknowledgements

We thank the Associate Editor and two anonymous reviewers for thoughtful comments leading us to add Remarks 1.2 and 1.3 and to develop a data-driven bandwidth selection method. A. Schick was supported in part by NSF Grant DMS 0405791.