Rates of the Strong Uniform Consistency with Rates for Conditional U-Statistics Estimators with General Kernels on Manifolds

IF 0.8 Q3 STATISTICS & PROBABILITY Mathematical Methods of Statistics Pub Date : 2024-07-15 DOI:10.3103/s1066530724700066

Salim Bouzebda, Nourelhouda Taachouche

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Abstract

$U$-statistics represent a fundamental class of statistics from modeling quantities of interest defined by multi-subject responses. $U$-statistics generalize the empirical mean of a random variable $X$ to sums over every $m$-tuple of distinct observations of $X$. Stute [103] introduced a class of so-called conditional $U$-statistics, which may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to:

$$r^{(k)}(\varphi,\tilde{\mathbf{t}}):=\mathbb{E}[\varphi(Y_{1},\ldots,Y_{k})|(X_{1},\ldots,X_{k})=\tilde{\mathbf{t}}]\quad\textrm{for}\quad\tilde{\mathbf{t}}=\left(\mathbf{t}_{1},\ldots,\mathbf{t}_{k}\right)\in\mathbb{R}^{dk}.$$

In the analysis of modern machine learning algorithms, sometimes we need to manipulate kernel estimation within the nonconventional setting with intricate kernels that might even be irregular and asymmetric. In this general setting, we obtain the strong uniform consistency result for the general kernel on Riemannian manifolds with Riemann integrable kernels for the conditional $U$-processes. We treat both cases when the class of functions is bounded or unbounded, satisfying some moment conditions. These results are proved under some standard structural conditions on the classes of functions and some mild conditions on the model. Our findings are applied to the regression function, the set indexed conditional $U$-statistics, the generalized $U$-statistics, and the discrimination problem. The theoretical results established in this paper are (or will be) key tools for many further developments in manifold data analysis.

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曲面上具有一般核的条件 U 统计估计器的强均匀一致率与率

Abstract$U$-statistics 代表了由多受试者反应定义的感兴趣数量建模的一类基本统计。$U$-statistics 将随机变量 $X$ 的经验平均值概括为 $X$ 的每一个 $m$-tuple 的不同观测值的总和。Stute[103]引入了一类所谓的条件（U）统计量，可以将其视为回归函数的 Nadaraya-Watson 估计值的一般化。Stute 证明了它们的强点一致性：$$r^{(k)}(\varphi,\tilde{\mathbf{t}})：=\mathbb{E}[\varphi(Y_{1},\ldots,Y_{k})|(X_{1},\ldots,X_{k})=\tilde{\mathbf{t}}]\quad\textrm{for}\quad\tilde{\mathbf{t}}=\left(\mathbf{t}_{1},\ldots,\mathbf{t}_{k}\right)\in\mathbb{R}^{dk}.$$ 在分析现代机器学习算法时，有时我们需要在非常规环境下使用错综复杂的内核来处理内核估计，这些内核甚至可能是不规则和不对称的。在这种一般情况下，我们得到了在黎曼流形上具有黎曼可积分核的条件（U\）过程的一般核的强均匀一致性结果。我们处理了满足某些矩条件的有界或无界函数类的两种情况。这些结果是在函数类的一些标准结构条件和模型的一些温和条件下证明的。我们的发现被应用于回归函数、集合索引条件 \（U\）统计量、广义 \（U\）统计量和判别问题。本文建立的理论结果是（或将是）流形数据分析进一步发展的关键工具。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Methods of Statistics STATISTICS & PROBABILITY-

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0.60

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0.00%

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期刊介绍： Mathematical Methods of Statistics is an is an international peer reviewed journal dedicated to the mathematical foundations of statistical theory. It primarily publishes research papers with complete proofs and, occasionally, review papers on particular problems of statistics. Papers dealing with applications of statistics are also published if they contain new theoretical developments to the underlying statistical methods. The journal provides an outlet for research in advanced statistical methodology and for studies where such methodology is effectively used or which stimulate its further development.