Rate of uniform consistency for nonparametric estimates with functional variables

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• 作者：Fré ; dé ; ric Ferraty ; Ali Laksaci ; Amel Tadj ; Philippe Vieu
• 关键词：Uniform almost complete convergence ; Kernel estimators ; Functional data ; Entropy ; Semi-metric space
• 刊名：Journal of Statistical Planning and Inference
• 出版年：2010
• 出版时间：February 2010
• 年：2010
• 卷：140
• 期：2
• 页码：335-352
• 全文大小：249 K

文摘

In this paper we investigate nonparametric estimation of some functionals of the conditional distribution of a scalar response variable Y given a random variable X taking values in a semi-metric space. These functionals include the regression function, the conditional cumulative distribution, the conditional density and some other ones. The literature on nonparametric functional statistics is only concerning pointwise consistency results, and our main aim is to prove the uniform almost complete convergence (with rate) of the kernel estimators of these nonparametric models. Unlike in standard multivariate cases, the gap between pointwise and uniform results is not immediate. So, suitable topological considerations are needed, implying changes in the rates of convergence which are quantified by entropy considerations. These theoretical uniform consistency results are (or will be) key tools for many further developments in functional data analysis.