The delta method for analytic functions of random operators with application to functional data

TL;DR

This paper develops a delta method for deriving the asymptotic distributions of estimators related to regularized functional canonical correlation, using eigenvalue perturbation theory for operators.

Contribution

It introduces a novel delta method for analytic functions of random operators, enabling asymptotic analysis of functional data estimators.

Findings

Derived the asymptotic distributions of regularized functional canonical correlation estimators.

Established weak convergence results for operators based on covariance operator convergence.

Applied perturbation theory to obtain limiting distributions of canonical quantities.

Abstract

In this paper, the asymptotic distributions of estimators for the regularized functional canonical correlation and variates of the population are derived. The method is based on the possibility of expressing these regularized quantities as the maximum eigenvalue and the corresponding eigenfunctions of an associated pair of regularized operators, similar to the Euclidean case. The known weak convergence of the sample covariance operator, coupled with a delta-method for analytic functions of covariance operators, yields the weak convergence of the pair of associated operators. From the latter weak convergence, the limiting distributions of the canonical quantities of interest can be derived with the help of some further perturbation theory.