Asymptotic equivalence of functional linear regression and a white noise inverse problem

TL;DR

This paper establishes the asymptotic equivalence between functional linear regression and a white noise inverse problem, enabling new theoretical insights and practical applications in statistical analysis of functional data.

Contribution

It introduces a white noise model for FLR and proves their asymptotic equivalence, extending results to finite samples and unknown design distributions.

Findings

Asymptotic equivalence of FLR and white noise model in LeCam's sense

Finite sample equivalence of FLR and empirical white noise model

Derivation of sharp minimax constants for FLR

Abstract

We consider the statistical experiment of functional linear regression (FLR). Furthermore, we introduce a white noise model where one observes an Ito process, which contains the covariance operator of the corresponding FLR model in its construction. We prove asymptotic equivalence of FLR and this white noise model in LeCam's sense under known design distribution. Moreover, we show equivalence of FLR and an empirical version of the white noise model for finite sample sizes. As an application, we derive sharp minimax constants in the FLR model which are still valid in the case of unknown design distribution.