Gaussian and bootstrap approximations for functional principal component regression

TL;DR

This paper demonstrates that, with proper operator scaling, Gaussian and bootstrap approximations are valid for functional principal component regression estimators, enabling reliable statistical inference.

Contribution

It establishes the first valid Gaussian and bootstrap approximation results for FPCR estimators under operator scaling, facilitating hypothesis testing in functional regression.

Findings

Valid Gaussian and bootstrap approximations hold for FPCR under operator scaling

The proposed methods perform well in hypothesis testing scenarios

Results enable new inferential tools for complex functional regression models

Abstract

Asymptotic inference using functional principal component regression (FPCR) has long been considered difficult, largely because, upon any scalar scaling, the FPCR estimator fails to satisfy a central limit theorem, leading to the prevailing belief that it is unsuitable for direct statistical inference. In this paper, we upend this traditional viewpoint by establishing a new result: upon suitable operator scaling, valid Gaussian and bootstrap approximations hold for the FPCR estimator. We apply this surprising finding to hypothesis testing for the significance of the slope function in functional regression models and demonstrate the strong numerical performance of the resulting tests. While concise, our results yield powerful inferential tools for functional regression. We believe it paves the way for new lines of inferential methodology for more complex functional regression settings.