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Gaussian and bootstrap approximations for functional principal component regression

Hyemin Yeon

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.

Gaussian and bootstrap approximations for functional principal component regression

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.
Paper Structure (16 sections, 17 theorems, 78 equations, 1 figure)

This paper contains 16 sections, 17 theorems, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Challenges
  4. Contributions
  5. Organization of the paper
  6. Main results
  7. SoFR models
  8. FPCR estimator and residual bootstrap
  9. Gaussian and bootstrap approximation results
  10. Statistical application to hypothesis testing for the significance of the slope function
  11. Conclusion
  12. Preliminaries
  13. Proof of the theorems
  14. Proofs for the variance terms
  15. Proofs for the bias term
  16. ...and 1 more sections

Key Result

Theorem 2.1

Suppose that Conditions condMomentX--condBasicTrunc and condErr hold along with Conditions condSlope for $v>2$ and condTrunc for $w > 6$. Then, the sampling distribution of $T_J$ is approximated by either a Gaussian or the bootstrap distributions as where $G_J$ is the Gaussian random element taking values in $\mathbb{H}$ with mean zero and covariance operator $\Pi_J \equiv \sum_{j=1}^J \phi_j^{\o

Figures (1)

  • Figure 3.1: Empirical rejection rates of the bootstrap tests using statistics $S_{\mathrm{sq},J}$ and $S_{\mathrm{sup},J}$ in \ref{['eqStatTest']}.

Theorems & Definitions (33)

  • Theorem 2.1
  • Remark
  • Theorem 3.1
  • Proposition A.1
  • proof
  • Proposition B.1
  • Proposition B.2
  • Proposition B.3
  • Proposition B.4
  • proof : Proof of \ref{['thmMain']}
  • ...and 23 more