Functional Uniform-in-Bandwidth Moderate Deviation Principle for the Local Empirical Processes Involving Functional Data

Our research employs general empirical process methods to investigate and establish moderate deviation principles for kernel-type function estimators that rely on an infinite-dimensional covariate, subject to mild regularity conditions. In doing so, we introduce a valuable moderate deviation principle for a function-indexed process, utilizing intricate exponential contiguity arguments. The primary objective of this paper is to contribute to the existing literature on functional data analysis by establishing functional moderate deviation principles for both Nadaraya–Watson and conditional distribution processes. These principles serve as fundamental tools for analyzing and understanding the behavior of these processes in the context of functional data analysis. By extending the scope of moderate deviation principles to the realm of functional data analysis, we enhance our understanding of the statistical properties and limitations of kernel-type function estimators when dealing with infinite-dimensional covariates. Our findings provide valuable insights and contribute to the advancement of statistical methodology in functional data analysis.