Nonparametric estimation when data on derivatives are available

• 成果类型: Article
• 署名作者: Hall, Peter; Yatchew, Adonis
• 署名单位: University of Melbourne; University of Toronto
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053606000001127
• 发表日期: 2007
• 页码: 300-323
• 关键词: projection pursuit; ADDITIVE-MODELS; regression; CONVERGENCE; rates

摘要

We consider settings where data are available on a nonparametric function and various partial derivatives. Such circumstances arise in practice, for example in the joint estimation of cost and input functions in economics. We show that when derivative data are available, local averages can be replaced in certain dimensions by nonlocal averages, thus reducing the nonparametric dimension of the problem. We derive optimal rates of convergence and conditions under which dimension reduction is achieved. Kernel estimators and their properties are analyzed, although other estimators, such as local polynomial, spline and nonparametric least squares, may also be used. Simulations and an application to the estimation of electricity distribution costs are included.