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Adaptive Global Testing for Functional Linear Models

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  • Jing Lei

Abstract

This article studies global testing of the slope function in functional linear regression models. A major challenge in functional global testing is to choose the dimension of projection when approximating the functional regression model by a finite dimensional multivariate linear regression model. We develop a new method that simultaneously tests the slope vectors in a sequence of functional principal components regression models. The sequence of models being tested is determined by the sample size and is an integral part of the testing procedure. Our theoretical analysis shows that the proposed method is uniformly powerful over a class of smooth alternatives when the signal to noise ratio exceeds the detection boundary. The methods and results reflect the deep connection between the functional linear regression model and the Gaussian sequence model. We also present an extensive simulation study and a real data example to illustrate the finite sample performance of our method. Supplementary materials for this article are available online.

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  • Jing Lei, 2014. "Adaptive Global Testing for Functional Linear Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(506), pages 624-634, June.
    • Handle: RePEc:taf:jnlasa:v:109:y:2014:i:506:p:624-634
    DOI: 10.1080/01621459.2013.856794
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    Cited by:

    1. Lai, Tingyu & Zhang, Zhongzhan & Wang, Yafei, 2021. "A kernel-based measure for conditional mean dependence," Computational Statistics & Data Analysis, Elsevier, vol. 160(C).
    2. Xu, Jianjun & Cui, Wenquan, 2022. "A new RKHS-based global testing for functional linear model," Statistics & Probability Letters, Elsevier, vol. 182(C).
    3. Zhang, Xiaochen & Zhang, Qingzhao & Ma, Shuangge & Fang, Kuangnan, 2022. "Subgroup analysis for high-dimensional functional regression," Journal of Multivariate Analysis, Elsevier, vol. 192(C).
    4. Christoph Breunig & Xiaohong Chen, 2020. "Adaptive, Rate-Optimal Hypothesis Testing in Nonparametric IV Models," Papers 2006.09587, arXiv.org, revised Nov 2024.
    5. Zhiqiang Jiang & Zhensheng Huang & Jing Zhang, 2023. "Functional single-index composite quantile regression," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(5), pages 595-603, July.
    6. Liu, Yuzi & Peng, Ling & Liu, Qing & Lian, Heng & Liu, Xiaohui, 2023. "Functional additive expectile regression in the reproducing kernel Hilbert space," Journal of Multivariate Analysis, Elsevier, vol. 198(C).
    7. Philip T. Reiss & Jeff Goldsmith & Han Lin Shang & R. Todd Ogden, 2017. "Methods for Scalar-on-Function Regression," International Statistical Review, International Statistical Institute, vol. 85(2), pages 228-249, August.
    8. Jadhav, Sneha & Ma, Shuangge, 2021. "An association test for functional data based on Kendall’s Tau," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    9. Xu, Wenchao & Zhang, Xinyu & Liang, Hua, 2024. "Linearized maximum rank correlation estimation when covariates are functional," Journal of Multivariate Analysis, Elsevier, vol. 202(C).

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