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

Abstract

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.

APPENDIX A

This appendix contains supplementary information that is essential to providing a more comprehensive understanding of the paper.

Theorem A.1 (Theorem 3.1 [6]). Let $\left\{U_{n}(t):t\in T\right\}$ be a sequence of stochastic processes, where $T$ is an index set. Let $\left\{\varepsilon_{n}\right\}$ be a sequence of positive numbers that converge to zero. Let $I:l_{\infty}(T)\rightarrow[0,\infty]$ and let $I_{t_{1},\ldots,t_{m}}:\mathbb{R}^{m}\rightarrow[0,\infty]$ be a function for each $t_{1},\ldots,t_{m}\in T$. Let $d(\cdot,\cdot)$ be a pseudometric in $T$. Consider the following conditions:

(a.1) $(T,d)$ is totally bounded;

(a.2) for each $t_{1},\ldots,t_{m}\in T,\left(U_{n}\left(t_{1}\right),\ldots,U_{n}\left(t_{m}\right)\right)$ satisfies the LDP with the rate $\varepsilon_{n}^{-1}$ and good rate function $I_{t_{1},\ldots,t_{m}}$;

(a.3) for each $\tau>0$,

$$\lim_{\eta\rightarrow 0}\limsup_{n\rightarrow\infty}\varepsilon_{n}\log\left(\mathbb{P}^{*}\left\{\sup_{d(s,t)\leq\eta}\left|U_{n}(t)-U_{n}(s)\right|\geq\tau\right\}\right)=-\infty;$$

(b.1) for each $0\leq c<\infty,\bigg{\{}z\in l_{\infty}(T):I(z)\leq c\bigg{\}}$ is a compact set of $l_{\infty}(T)$;

(b.2) for each $A\subset l_{\infty}(T)$,

$$-\inf_{z\in A^{\circ}}I(z)\leq\liminf_{n\rightarrow\infty}\varepsilon_{n}\log\left(\mathbb{P}_{*}\left\{U_{n}\in A\right\}\right)$$

$${}\leq\limsup_{n\rightarrow\infty}\varepsilon_{n}\log\left(\mathbb{P}^{*}\left\{U_{n}\in A\right\}\right)\leq-\inf_{z\in\bar{A}}I(z).$$

If the set of conditions (a) is satisfied, then the set of conditions (b) holds with $I(\cdot)$ given by

$$I(z)=\sup\bigg{\{}I_{t_{1},\ldots,t_{m}}\left(z\left(t_{1}\right),\ldots,z\left(t_{m}\right)\right):t_{1},\ldots,t_{m}\in T,m\geq 1\bigg{\}}.$$

If the set of conditions (b) is satisfied, then the set of conditions (a) holds with

$$I_{t_{1},\ldots,t_{m}}\left(u_{1},\ldots,u_{m}\right)=\inf\bigg{\{}I(z):z\in l_{\infty}(T),\left(z\left(t_{1}\right),\ldots,z\left(t_{m}\right)\right)=\left(u_{1},\ldots,u_{m}\right)\bigg{\}}$$

and the pseudometric $d(\cdot,\cdot)$ is defined

$$d(s,t)=\sum_{k=1}^{\infty}k^{-2}\min\left(d_{k}(s,t),1\right),$$

where

$$d_{k}(s,t)=\sup\left\{\left|u_{2}-u_{1}\right|:I_{s,t}\left(u_{1},u_{2}\right)\leqq k\right\}.$$