Uniform consistency and uniform in bandwidth consistency for nonparametric regression estimates and conditional U-statistics involving functional data

被引:43
作者
Bouzebda, Salim [1 ]
Nemouchi, Boutheina [1 ]
机构
[1] Univ Technol Compiegne, Alliance Sorbonne Univ, LMAC, Compiegne, France
关键词
Uniform almost complete convergence; conditional U-statistic; functional data analysis; functional regression; Kolmogorov's entropy; small ball probability; uniform consistency; uniform in bandwidth consistency; Kernel-type estimators; LIMIT-THEOREMS; KERNEL ESTIMATORS; DENSITY; RANKING; ERROR; LAWS;
D O I
10.1080/10485252.2020.1759597
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
W. Stute [(1991), Annals of Probability, 19, 812-825] introduced a class of so-called conditional U-statistics, which may be viewed as a generalisation of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency toWe apply the methods developed in Dony and Mason [(2008), Bernoulli, 14(4), 1108-1133] to establish uniformity in and in bandwidth consistency (i.e. , where at some specific rate) to of the estimator proposed by Stute when Y and covariates X are functional taking value in some abstract spaces. In addition, uniform consistency is also established over for a suitably restricted class . The theoretical uniform consistency results, established in this paper, are (or will be) key tools for many further developments in functional data analysis. Applications include the Nadaraya-Watson kernel estimators and the conditional distribution function. Our theorems allow data-driven local bandwidths for these statistics.
引用
收藏
页码:452 / 509
页数:58
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