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Inference for function-on-function regression: central limit theorem and residual bootstrap

Hyemin Yeon

Abstract

We investigate asymptotic inference in a linear regression model where both response and regressors are functions, using an estimator based on functional principal components analysis. Although this approach is widely used in functional data analysis, there remains significant room for developing its asymptotic properties for function-on-function regression. Our study targets the mean response at a new regressor with two primary aims. First, we refine the existing central limit theorem by relaxing certain technical conditions, which include generalizing the scaling factor, resulting in incorporating a broader class of random functions beyond those having scores with independence or finite higher moments. Second, we introduce a residual bootstrap method that enhances the calibration of various confidence sets for quantities related to mean response, while its consistency is rigorously verified. Numerical studies compare the finite sample performance of both asymptotic and bootstrap approaches, demonstrating higher accuracy of the latter. To illustrate bootstrap inference for mean response, we apply it to the Canadian weather dataset.

Inference for function-on-function regression: central limit theorem and residual bootstrap

Abstract

We investigate asymptotic inference in a linear regression model where both response and regressors are functions, using an estimator based on functional principal components analysis. Although this approach is widely used in functional data analysis, there remains significant room for developing its asymptotic properties for function-on-function regression. Our study targets the mean response at a new regressor with two primary aims. First, we refine the existing central limit theorem by relaxing certain technical conditions, which include generalizing the scaling factor, resulting in incorporating a broader class of random functions beyond those having scores with independence or finite higher moments. Second, we introduce a residual bootstrap method that enhances the calibration of various confidence sets for quantities related to mean response, while its consistency is rigorously verified. Numerical studies compare the finite sample performance of both asymptotic and bootstrap approaches, demonstrating higher accuracy of the latter. To illustrate bootstrap inference for mean response, we apply it to the Canadian weather dataset.
Paper Structure (13 sections, 4 theorems, 43 equations, 1 figure, 2 tables)

This paper contains 13 sections, 4 theorems, 43 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Estimation for linear FoFR
  4. CLT for mean response
  5. Assumptions
  6. Generalized CLT
  7. Residual bootstrap
  8. Numerical studies
  9. Asymptotic and bootstrap confidence sets
  10. Simulation setups
  11. Empirical coverage rates
  12. Real data example
  13. Conclusions and outlook

Key Result

Theorem 1

Suppose Conditions condModel-condBasicTrunc and condError hold along with Condition condScale and We additionally suppose that Condition condBias holds for $u>5$ along with where the function $m$ is given in (eq_mju). Then, the mean response statistic $T_n$ in (eqStat) weakly converges to the centered Gaussian distribution $\mathsf{G}(0,\Sigma)$ with covariance $\Sigma$ conditionally on $\mathca

Figures (1)

  • Figure 1: Canadian weather dataset. Averaged daily temperatures (upper) and (log) precipitations (lower) over a year for four regions in Canada, where the regional average curves are colored in blue.

Theorems & Definitions (5)

  • Example 1
  • Theorem 1
  • Corollary 1
  • Proposition 1
  • Theorem 2