Large Deviation Results for the Nonparametric Regression Function Estimator on Functional Data

TL;DR

This paper investigates the large deviation properties of a nonparametric regression estimator for functional data, establishing principles that describe the probabilities of significant deviations in estimation accuracy.

Contribution

It introduces a large deviation principle for a process related to the estimator and derives new pointwise and uniform large deviation results for the regression estimator.

Findings

Large deviation principle for the process Zn.

Pointwise large deviation principle for the estimator.

Uniform large deviation result over VC-classes.

Abstract

This paper is devoted to the study of large deviation behaviors in the setting of the estimation of the regression function on functional data. A large deviation principle is stated for a process Zn, defined below, allowing to derive a pointwise large deviation principle for the Nadaraya-Watson-type l-indexed regression function estimator as a by-product. Moreover, a uniform over VC-classes Cherno? type large deviation result is stated for the deviation of the l-indexed regression estimator.