MULTIVARIATE LOCALLY WEIGHTED LEAST-SQUARES REGRESSION

• 成果类型: Article
• 署名作者: RUPPERT, D; WAND, MP
• 署名单位: University of New South Wales Sydney
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176325632
• 发表日期: 1994
• 页码: 1346-1370
• 关键词: Nonparametric regression; CONVERGENCE; smoothers; rates

摘要

Nonparametric regression using locally weighted least squares was first discussed by Stone and by Cleveland. Recently, it was shown by Fan and by Fan and Gijbels that the local linear kernel-weighted least squares regression estimator has asymptotic properties making it superior, in certain senses, to the Nadaraya-Watson and Gasser-Muller kernel estimators. In this paper we extend their results on asymptotic bias and variance to the case of multivariate predictor variables. We are able to derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory. This approach is especially convenient when analyzing the asymptotic conditional bias and variance of the estimator at points near the boundary of the support of the predictors. We also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator discussed by Cleveland and Devlin and, in the univariate case, higher-order polynomial fits and derivative estimation.