Variance Function Estimation in Regression: The Effect of Estimating the Mean
Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 1989 · Volume 51 · Issue 1
DOI: 10.1111/j.2517-6161.1989.tb01744.x
Abstract
We consider estimation of a variance functiongin regression problems. Such estimation requires simultaneous estimation of the mean functionf. We obtain clear results on the extent to which the smoothness offinfluences best rates of convergence for estimatingg.For example, in nonparametric regression with two derivatives ong, `classical' rates of convergence are possible if and only if the unknownfsatisfies a Lipschitz condition of orderor more. If a parametric model is known forg, thengmay be estimatedconsistently if and only iffis Lipschitz of orderor more. Optimal rates of convergence are attained by Kernel estimators.
Keywords
heteroscedasticity, nonparametric regression, rates of convergence, variance functions
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