Variance Function Estimation in Regression: The Effect of Estimating the Mean

作者：Peter M. Hall, Raymond James Carroll · 发表于：Journal of the Royal Statistical Society Series B (Statistical Methodology) · 年份：1989 · DOI：10.1111/j.2517-6161.1989.tb01744.x · 被引用次数：160 · 研究领域：Statistical Methods and Inference、Stochastic Gradient Optimization Techniques、Gaussian Processes and Bayesian Inference

SUMMARY We consider estimation of a variance function g in regression problems. Such estimation requires simultaneous estimation of the mean function f. We obtain clear results on the extent to which the smoothness of f influences best rates of convergence for estimating g. For example, in nonparametric regression with two derivatives on g, ‘classical' rates of convergence are possible if and only if the unknown f satisfies a Lipschitz condition of order 13 or more. If a parametric model is known for g, then g may be estimated n1/2 consistently if and only if f is Lipschitz of order 12 or more. Optimal rates of convergence are attained by Kernel estimators.