Uniform limit theorems for a class of conditional [formula omitted]-estimators when covariates are functions

Abstract

This paper considers nonparametric estimation of a parameter, which is a zero of a certain estimating equation, indexed by a class of functions and depending on an infinite-dimensional covariate. This problem is known in the literature as Z-estimation, where a rich theory, in the case of the finite-dimensional covariates, exists. However, in certain applications, covariates may belong to an infinite-dimensional space, which makes the estimation as well as the study of the asymptotic properties challenging topics. This paper aims to bring several contributions to the existing functional data analysis literature. First, we introduce nonparametric conditional Z-estimators whenever functional stationary ergodic data are observed under right-censorship model. A sharp uniform-in-bandwidth limit law for the proposed estimator is presented. Such result allows the bandwidth 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. Moreover, a uniform consistency rate, over Vapnik–Chervonenkis classes of functions, is provided for the process of conditional Z-estimators. Finally, a uniform central limit theorem for the process indexed by functions is established. The last result is of particular interest since it is obtained for dependent functional data. The uniform limit theorems, discussed in this paper, are key tools for many further developments in functional data analysis involving empirical process techniques. These results are proved under some standard structural conditions on the Vapnik–Chervonenkis classes of functions and some mild conditions on the model.