Uniform in number of neighbors consistency and weak convergence of kNN empirical conditional processes and kNN conditional U-processes involving functional mixing data

Abstract

$ U $-statistics represent a fundamental class of statistics arising 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 [182] 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^{(m)}(\varphi, \mathbf{t}): = \mathbb{E}[\varphi(Y_{1}, \ldots, Y_{m})|(X_{1}, \ldots, X_{m}) = \mathbf{t}], \; \mbox{for}\; \mathbf{ t}\in \mathcal{X}^{m}. $ In this paper, we are mainly interested in the study of the $ k $NN conditional $ U $-processes in a functional mixing data framework. More precisely, we investigate the weak convergence of the conditional empirical process indexed by a suitable class of functions and of the $ k $NN conditional $ U $-processes when the explicative variable is functional. We treat the uniform central limit theorem in both cases when the class of functions is bounded or unbounded satisfying some moment conditions. The second main contribution of this study is the establishment of a sharp almost complete Uniform consistency in the Number of Neighbors of the constructed estimator. Such a result allows the number of neighbors to vary within a complete range for which the estimator is consistent. Consequently, it represents an interesting guideline in practice to select the optimal bandwidth in nonparametric functional data analysis. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis classes of functions and some mild conditions on the model. The theoretical results established in this paper are (or will be) key tools for further functional data analysis developments. Potential applications include the set indexed conditional U-statistics, Kendall rank correlation coefficient, the discrimination problems and the time series prediction from a continuous set of past values.

Keywords

• conditional $ u $-statistic
• functional data analysis
• functional regression
• kolmogorov's entropy
• small ball probability
• $ k $nn kernel-type estimators
• empirical processes
• $ u $-processes
• weak convergence
• vc-classes