Abstract

Functional regressors complicate inference in linear regression problems so that the

bootstrap can play a useful role in quantifying uncertainty and calibrating intervals. The best

bootstrap in practice, though, can depend on factors in the data as well as computational considerations and existing bootstraps can have limitations: residual bootstrap is computationally

fast and simple but may fail when the errors are heterogeneous, while paired bootstrap applies

more generally in functional linear regression at a cost of much higher computation. To bridge

this gap, we develop a wild bootstrap method for functional linear regression, which is akin to

a modified version of residual bootstrap but designed to have a wide scope of application like

paired bootstrap, including to heteroscedastic errors. Its theoretical consistency is established

and numerical studies suggest that wild bootstrap can provide accurate and computationally fast

inference. Importantly, we also suggest a practical and effective approach of selecting truncation levels, specifically designed for mean response inference problems. The proposed bootstrap

in functional linear regression is further illustrated through a weather data example, and an

accompanying R package BTSinFLRM provides numerical implementations.

Key words and phrases: functional principal components analysis, heteroscedasticity, infinite dimensionality, resampling, scalar-on-function regression, stabilized volatility method 1

Information

Preprint No.SS-2025-0307
Manuscript IDSS-2025-0307
Complete AuthorsHyemin Yeon, Xiongtao Dai, Daniel Nordman
Corresponding AuthorsHyemin Yeon
Emailshyeon2@gmu.edu

References

  1. Cai, T. T. and P. Hall (2006). Prediction in functional linear regression. Annals of Statistics 34(5), 2159–2179.
  2. Cai, T. T., L. Zhang, and H. H. Zhou (2018). Adaptive functional linear regression via functional principal component analysis and block thresholding. Statistica Sinica 28(4), 2455–2468.
  3. Cao-Abad, R. (1991). Rate of convergence for the wild bootstrap in nonparametric regression. Annals of Statistics 19(4), 2226–2231.
  4. Cardot, H., F. Ferraty, and P. Sarda (1999). Functional linear model. Statistics & Probability Letters 45(1), 11–22.
  5. Cardot, H., F. Ferraty, and P. Sarda (2003). Spline estimators for the functional linear model. Statistica Sinica, 571–591.
  6. Cardot, H., A. Mas, and P. Sarda (2007). CLT in functional linear regression models. Probability Theory and Related Fields 138(3-4), 325–361.
  7. Dette, H. and J. Tang (2024). Statistical inference for function-on-function linear regression. Bernoulli 30(1), 304–331.
  8. Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Annals of Statistics 7(1), 1–26.
  9. Freedman, D. A. (1981). Bootstrapping regression models. Annals of Statistics 9(6), 1218–1228.
  10. Gonz´alez-Manteiga, W. and A. Mart´ınez-Calvo (2011). Bootstrap in functional linear regression. Journal of Statistical Planning and Inference 141(1), 453–461.
  11. Hall, P. (1988). On symmetric bootstrap confidence intervals. Journal of the Royal Statistical Society Series B: Statistical Methodology 50(1), 35–45.
  12. Hall, P. and J. L. Horowitz (2007). Methodology and convergence rates for functional linear regression. Annals of Statistics 35(1), 70–91.
  13. H¨ardle, W. and E. Mammen (1991). Bootstrap methods in nonparametric regression. In Nonparametric Functional Estimation and Related Topics, pp. 111–123. Springer.
  14. H¨ardle, W. and J. S. Marron (1991). Bootstrap simultaneous error bars for nonparametric regression. Annals of Statistics, 778–796.
  15. Hastie, T., R. Tibshirani, and J. Friedman (2009). The Elements of Statistical Learning. Springer New York.
  16. Hsing, T. and R. Eubank (2015). Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators, Volume 997. John Wiley & Sons.
  17. Khademnoe, O. and S. M. E. Hosseini-Nasab (2024). On the validity of the bootstrap hypothesis testing in functional linear regression. Statistical Papers 65(4), 2361–2396.
  18. Kokoszka, P. and M. Reimherr (2017). Introduction to Functional Data Analysis. Chapman and Hall/CRC.
  19. Lin, Y. and Z. Lin (2025). Hypothesis testing for functional linear models via bootstrapping. Bernoulli 31(4), 3309–3330.
  20. Liu, R. Y. (1988). Bootstrap procedures under some non-iid models. Annals of Statistics 16(4), 1696–1708.
  21. Mammen, E. (1993). Bootstrap and wild bootstrap for high dimensional linear models. Annals of Statistics 21(1), 255–285.
  22. Mammen, E. (2000). Resampling methods for nonparametric regression. Smoothing and Regression: Approaches, Computation, and Application, 425–450.
  23. Mersmann, O. (2025). microbenchmark: Accurate Timing Functions. R package version 1.5.0.
  24. M¨uller, H.-G. and U. Stadtm¨uller (2005). Generalized functional linear models. Annals of Statistics 33(2), 774–805.
  25. Politis, D., J. Romano, and M. Wolf (1999). Subsampling. Springer Series in Statistics. Springer New York.
  26. Shin, H. (2009). Partial functional linear regression. Journal of Statistical Planning and Inference 139(10), 3405–3418.
  27. Wu, C.-F. J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis. Annals of Statistics 14(4), 1261–1295.
  28. Xue, K. and F. Yao (2021). Hypothesis testing in large-scale functional linear regression. Statistica Sinica 31(2), 1101–1123.
  29. Yeon, H. (2026). Gaussian and bootstrap approximation for functional principal component regression. To appear in Electronical Journal of Statistics.
  30. Yeon, H., X. Dai, and D. J. Nordman (2023). Bootstrap inference in functional linear regression models with scalar response. Bernoulli 29(4), 2599 – 2626.
  31. Yeon, H., X. Dai, and D. J. Nordman (2024a). Bootstrap inference in functional linear regression models with scalar response under heteroscedasticity. Electronic Journal of Statistics 18(2), 3590 – 3627.
  32. Yeon, H., X. Dai, and D. J. Nordman (2024b).
  33. Supplement to “bootstrap inference in functional linear regression models with scalar response under heteroscedasticity”. Iowa State University Digital Repository, .
  34. Zhou, H., F. Yao, and H. Zhang (2023). Functional linear regression for discretely observed data: from ideal to reality. Biometrika 110(2), 381–393.

Acknowledgments

We are thankful to associate editor and two anonymous reviewers for their constructive

and helpful feedback. A large part of the work by Hyemin Yeon was carried out while he

was at Kent State University, and research is partially supported by the Department

of Mathematical Sciences at Kent State University (Yeon) and NSF DMS-2515719

(Nordman).

Supplementary Materials

This supplement contains all technical details, additional simulation results, and extra

figures regarding data example.

Supplementary materials are available for download.