IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i8p1812-d1120640.html
   My bibliography  Save this article

An Adaptive-to-Model Test for Parametric Functional Single-Index Model

Author

Listed:
  • Lili Xia

    (Faculty of Science, Beijing University of Technology, Beijing 100124, China)

  • Tingyu Lai

    (School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China)

  • Zhongzhan Zhang

    (Faculty of Science, Beijing University of Technology, Beijing 100124, China)

Abstract

Model checking methods based on non-parametric estimation are widely used because of their tractable limiting null distributions and being sensitive to high-frequency oscillation alternative models. However, this kind of test suffers from the curse of dimensionality, resulting in slow convergence, especially for functional data with infinite dimensional features. In this paper, we propose an adaptive-to-model test for a parametric functional single-index model by using the orthogonality of residual and its conditional expectation. The test achieves model adaptation by sufficient dimension reduction which utilizes functional sliced inverse regression. This test procedure can be easily extended to other non-parametric test methods. Under certain conditions, we prove the asymptotic properties of the test statistic under the null hypothesis, fixed alternative hypothesis and local alternative hypothesis. Simulations show that our test has better performance than the method that does not use functional sufficient dimension reduction. An analysis of COVID-19 data verifies our conclusion.

Suggested Citation

  • Lili Xia & Tingyu Lai & Zhongzhan Zhang, 2023. "An Adaptive-to-Model Test for Parametric Functional Single-Index Model," Mathematics, MDPI, vol. 11(8), pages 1-25, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:8:p:1812-:d:1120640
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/8/1812/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/8/1812/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Wang, Guochang & Lin, Nan & Zhang, Baoxue, 2014. "Functional k-means inverse regression," Computational Statistics & Data Analysis, Elsevier, vol. 70(C), pages 172-182.
    2. Mikosch, T., 1991. "Functional limit theorems for random quadratic forms," Stochastic Processes and their Applications, Elsevier, vol. 37(1), pages 81-98, February.
    3. Hervé Cardot & Frédéric Ferraty & André Mas & Pascal Sarda, 2003. "Testing Hypotheses in the Functional Linear Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(1), pages 241-255, March.
    4. Ellison, Glenn & Ellison, Sara Fisher, 2000. "A simple framework for nonparametric specification testing," Journal of Econometrics, Elsevier, vol. 96(1), pages 1-23, May.
    5. Valentin Patilea & César Sánchez-Sellero & Matthieu Saumard, 2016. "Testing the Predictor Effect on a Functional Response," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(516), pages 1684-1695, October.
    6. Shujie Ma, 2016. "Estimation and inference in functional single-index models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(1), pages 181-208, February.
    7. Zhu, Lixing & Miao, Baiqi & Peng, Heng, 2006. "On Sliced Inverse Regression With High-Dimensional Covariates," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 630-643, June.
    8. Cardot, Hervé & Sarda, Pacal, 2005. "Estimation in generalized linear models for functional data via penalized likelihood," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 24-41, January.
    9. Zheng, John Xu, 1998. "A Consistent Nonparametric Test Of Parametric Regression Models Under Conditional Quantile Restrictions," Econometric Theory, Cambridge University Press, vol. 14(1), pages 123-138, February.
    10. Winfried Stute & Li‐Xing Zhu, 2002. "Model Checks for Generalized Linear Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 29(3), pages 535-545, September.
    11. Zhu, Li-Ping & Zhu, Li-Xing & Feng, Zheng-Hui, 2010. "Dimension Reduction in Regressions Through Cumulative Slicing Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 105(492), pages 1455-1466.
    12. Liping Zhu & Tao Wang & Lixing Zhu & Louis Ferré, 2010. "Sufficient dimension reduction through discretization-expectation estimation," Biometrika, Biometrika Trust, vol. 97(2), pages 295-304.
    13. Guochang Wang & Xiang-Nan Feng & Min Chen, 2016. "Functional Partial Linear Single-index Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(1), pages 261-274, March.
    14. Xu Guo & Tao Wang & Lixing Zhu, 2016. "Model checking for parametric single-index models: a dimension reduction model-adaptive approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(5), pages 1013-1035, November.
    15. Zhang, Jia & Chen, Xin, 2019. "Robust sufficient dimension reduction via ball covariance," Computational Statistics & Data Analysis, Elsevier, vol. 140(C), pages 144-154.
    16. Lian, Heng & Li, Gaorong, 2014. "Series expansion for functional sufficient dimension reduction," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 150-165.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Xie, Chuanlong & Zhu, Lixing, 2019. "A goodness-of-fit test for variable-adjusted models," Computational Statistics & Data Analysis, Elsevier, vol. 138(C), pages 27-48.
    2. Wang, Qin & Xue, Yuan, 2021. "An ensemble of inverse moment estimators for sufficient dimension reduction," Computational Statistics & Data Analysis, Elsevier, vol. 161(C).
    3. Cuizhen Niu & Lixing Zhu, 2018. "A robust adaptive-to-model enhancement test for parametric single-index models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(5), pages 1013-1045, October.
    4. Chuanlong Xie & Lixing Zhu, 2018. "A minimum projected-distance test for parametric single-index Berkson models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(3), pages 700-715, September.
    5. Guochang Wang, 2017. "Dimension reduction in functional regression with categorical predictor," Computational Statistics, Springer, vol. 32(2), pages 585-609, June.
    6. Linjuan Zheng & Beiting Liang & Guochang Wang, 2024. "Adaptive slicing for functional slice inverse regression," Statistical Papers, Springer, vol. 65(5), pages 3261-3284, July.
    7. Deng, Jianqiu & Yang, Xiaojie & Wang, Qihua, 2022. "Surrogate space based dimension reduction for nonignorable nonresponse," Computational Statistics & Data Analysis, Elsevier, vol. 168(C).
    8. Chen, Feifei & Jiang, Qing & Feng, Zhenghui & Zhu, Lixing, 2020. "Model checks for functional linear regression models based on projected empirical processes," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    9. Zeng, Bilin & Yu, Zhou & Wen, Xuerong Meggie, 2015. "A note on cumulative mean estimation," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 322-327.
    10. Zhu, Xuehu & Guo, Xu & Wang, Tao & Zhu, Lixing, 2020. "Dimensionality determination: A thresholding double ridge ratio approach," Computational Statistics & Data Analysis, Elsevier, vol. 146(C).
    11. Xie, Chuanlong & Zhu, Lixing, 2020. "Generalized kernel-based inverse regression methods for sufficient dimension reduction," Computational Statistics & Data Analysis, Elsevier, vol. 150(C).
    12. Chen, Canyi & Xu, Wangli & Zhu, Liping, 2022. "Distributed estimation in heterogeneous reduced rank regression: With application to order determination in sufficient dimension reduction," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    13. Feng, Zhenghui & Zhu, Lixing, 2012. "An alternating determination–optimization approach for an additive multi-index model," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1981-1993.
    14. Guochang Wang & Jianjun Zhou & Wuqing Wu & Min Chen, 2017. "Robust functional sliced inverse regression," Statistical Papers, Springer, vol. 58(1), pages 227-245, March.
    15. Ming-Yueh Huang & Chin-Tsang Chiang, 2017. "An Effective Semiparametric Estimation Approach for the Sufficient Dimension Reduction Model," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1296-1310, July.
    16. Zhenghui Feng & Lu Lin & Ruoqing Zhu & Lixing Zhu, 2020. "Nonparametric variable selection and its application to additive models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(3), pages 827-854, June.
    17. Escanciano, Juan Carlos & Mayoral, Silvia, 2010. "Data-driven smooth tests for the martingale difference hypothesis," Computational Statistics & Data Analysis, Elsevier, vol. 54(8), pages 1983-1998, August.
    18. Hervé Cardot & Luboš Prchal & Pascal Sarda, 2007. "No effect and lack-of-fit permutation tests for functional regression," Computational Statistics, Springer, vol. 22(3), pages 371-390, September.
    19. Sudaraka Tholkage & Qi Zheng & Karunarathna B. Kulasekera, 2022. "Conditional Kaplan–Meier Estimator with Functional Covariates for Time-to-Event Data," Stats, MDPI, vol. 5(4), pages 1-17, November.
    20. Lian, Heng & Li, Gaorong, 2014. "Series expansion for functional sufficient dimension reduction," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 150-165.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:8:p:1812-:d:1120640. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.