ABSTRACT 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 to We 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.

中文翻译：

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涉及功能数据的非参数回归估计和条件 U 统计的均匀一致性和带宽一致性

摘要 W. Stute [(1991), Annals of Probability, 19, 812-825] 介绍了一类所谓的条件 U 统计，可以将其视为对回归函数的 Nadaraya-Watson 估计的概括。Stute 证明了它们强大的逐点一致性。我们应用 Dony 和 Mason [(2008), Bernoulli, 14(4), 1108–1133] 中开发的方法来建立带宽一致性（即，在某些特定速率下）的一致性当 Y 和协变量 X 是函数在某些抽象空间中取值时，Stute 提出的估计量。此外，还为适当限制的类建立了统一的一致性。本文中建立的理论统一一致性结果是（或将是）功能数据分析中许多进一步发展的关键工具。应用包括 Nadaraya-Watson 核估计器和条件分布函数。我们的定理允许数据驱动的本地带宽用于这些统计数据。