Universally consistent conditional \(U\)-statistics
type: Item
instance of: scholarly article
title: Universally consistent conditional \(U\)-statistics
zbMATH Open document ID: 0818.62049
published in: The Annals of Statistics
publication date: 1995-07-02T00:00:00Z
review text: This paper constitutes a continuation of work on so-called conditional (local) \(U\)-statistics as introduced in our paper, Ann. Probab. 19, No. 2, 812-825 (1991; Zbl 0770.60035). These statistics may be viewed as generalizations of regression function estimates, similar to Hoeffding's extension of a sample mean to what is now called a \(U\)-statistic. To be specific, assume that \((X_ i, Y_ i)\), \(1\leq i\leq n\), are i.i.d. random vectors in some Euclidean space \(\mathbb{R}^{d+s}\), defined on some probability space \((\Omega, {\mathcal A}, \mathbb{P})\). As in the regression case the \(X\) and \(Y\) may be considered as input and output variables, respectively, the stochastic dependence of which being described by the pertaining regression function. In order to measure the impact of a few \(X\)'s, say \((X_ 1,\dots, X_ k)\), on a function \(h(Y_ 1, \dots, Y_ k)\) of the pertaining \(Y\)'s, set \[ m({\mathbf x})\equiv m(x_ 1,\dots, x_ k):= \mathbb{E}[ h(Y_ 1, \dots, Y_ k)\mid X_ 1= x_ 1,\dots, X_ k= x_ k]. \] Here \(h\) is a given real-valued function on \(\mathbb{R}^{s\times k}\) (the \(U\)-kernel) such that, for some \(r\geq 1\), \(h(Y_ 1,\dots, Y_ k)\in {\mathcal L}_ r\), the space of all random variables \(Z\) for which \(| Z|^ r\) is integrable; \(k\) is called the degree of \(h\) (resp. \(m\)). We study a fairly general class of conditional \(U\)-statistics of the form \[ m_ n ({\mathbf x})= \sum_ \pi W_{\pi n} ({\mathbf x}) h(Y_ \pi), \tag{1} \] designed to estimate \(m({\mathbf x})\). Summation in (1) takes place over all permutations \(\pi= (\pi_ 1,\dots, \pi_ k)\) of length \(k\) such that \(1\leq \pi_ i\leq n\). Of course, \(Y_ \pi= (Y_{\pi_ 1}, \dots, Y_{\pi_ k})\). While in our previous paper pointwise consistency and asymptotic normality of \(m_ n\) were investigated, the present paper deals with consistency in \(r\)th mean. Theorem 1.1 presents sufficient conditions for universal consistency. In Theorems 1.2 and 1.3 it is shown that window weights…
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62G07
Mathematics Subject Classification ID: 62H30
zbMATH DE Number: 597422
zbMATH Keywords: nearest neighbor weights
zbMATH Keywords: nearest neighbor estimators
zbMATH Keywords: conditional \(U\)- statistics
zbMATH Keywords: sufficient conditions for universal consistency
zbMATH Keywords: window weights
zbMATH Keywords: nonparametric discrimination
author: Winfried Stute
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176325378
OpenAlex ID: W1970211457
DOI: 10.1214/AOS/1176325378
description: scientific article; zbMATH DE number 597422