Universally consistent conditional \(U\)-statistics
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 and \(k_ n\)-nearest neighbor (NN) weights are universally consistent under mild assumptions on the smoothing parameters. An application to a new nonparametric discrimination problem is also given.
sufficient conditions for universal consistency
scientific article; zbMATH DE number 597422