Relative Errors of Difference-Based Variance Estimators in Nonparametric Regression

Abstract

Difference-based estimators for the error variance are popular since they do not require the estimation of the mean function. Unlike most existing difference-based estimators, new estimators proposed by Müller et al. (2003) and Tong and Wang (2005) achieved the asymptotic optimal rate as residual-based estimators. In this article, we study the relative errors of these difference-based estimators which lead to better understanding of the differences between them and residual-based estimators. To compute the relative error of the covariate-matched U-statistic estimator proposed by Müller et al. (2003), we develop a modified version by using simpler weights. We further investigate its asymptotic property for both equidistant and random designs and show that our modified estimator is asymptotically efficient.

Keywords:

• Asymptotically efficient
• Bandwidth
• Kernel estimator
• Mean squared error
• Nonparametric regression
• Variance estimation

Mathematics Subject Classification:

• Primary 62G08
• Secondary 62G20

Acknowledgment

This research was supported by NIH Grant R01 GM58533.