Search Advanced

Electronic Research Archive

2026, Volume 34, Issue 1: 411-423. doi: 10.3934/era.2026019
Previous Article Next Article
Research article

A new measure of partial conditional mean independence in Hilbert spaces

  • School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100124, China
  • Published: 14 January 2026

  • A novel metric, called partial martingale difference-angle divergence, is proposed to test and measure partial conditional mean (in)dependence for Hilbert elements. The partial martingale difference-angle divergence has some appealing properties. It is nonnegative and equals zero if and only if the partial conditional mean independence holds; it has a simple expectation form; it does not require the moment condition for the predictor variable. We construct an estimator for partial martingale difference-angle divergence and derive its asymptotic properties. Finite sample simulations show that the proposed test performs well and has strong testing power for nonlinear relationships. Two real data examples are introduced to illustrate the application of the proposed test.

    Citation: Jiangyuan Bian, Zhongzhan Zhang. A new measure of partial conditional mean independence in Hilbert spaces[J]. Electronic Research Archive, 2026, 34(1): 411-423. doi: 10.3934/era.2026019

    Related Papers:

  • A novel metric, called partial martingale difference-angle divergence, is proposed to test and measure partial conditional mean (in)dependence for Hilbert elements. The partial martingale difference-angle divergence has some appealing properties. It is nonnegative and equals zero if and only if the partial conditional mean independence holds; it has a simple expectation form; it does not require the moment condition for the predictor variable. We construct an estimator for partial martingale difference-angle divergence and derive its asymptotic properties. Finite sample simulations show that the proposed test performs well and has strong testing power for nonlinear relationships. Two real data examples are introduced to illustrate the application of the proposed test.



    加载中


    [1] M. A. Delgado, W. G. Manteiga, Significance testing in nonparametric regression based on the bootstrap, Ann. Stat., 29 (2001), 1469–1507. https://doi.org/10.1214/aos/1013203462 doi: 10.1214/aos/1013203462
    [2] Y. Fan, Q. Li, Consistent model specification tests: omitted variables and semiparametric functional forms, Econometrica, 64 (1996), 865–890. https://doi.org/10.2307/2171848 doi: 10.2307/2171848
    [3] X. Shao, J. Zhang, Martingale difference correlation and its use in high-dimensional variable screening, J. Am. Stat. Assoc., 109 (2014), 1302–1318. https://doi.org/10.1080/01621459.2014.887012 doi: 10.1080/01621459.2014.887012
    [4] C. E. Lee, X. Zhang, X. Shao, Testing conditional mean independence for functional data, Biometrika, 107 (2020), 331–346. https://doi.org/10.1093/biomet/asz070 doi: 10.1093/biomet/asz070
    [5] T. Lai, Z. Zhang, Y. Wang, A kernel-based measure for conditional mean dependence, Comput. Stat. Data Anal., 160 (2021), 107246. https://doi.org/10.1016/j.csda.2021.107246 doi: 10.1016/j.csda.2021.107246
    [6] T. Lai, Z. Zhang, Local influence detection of conditional mean dependence, Commun. Math. Stat., 11 (2023), 1–27. https://doi.org/10.1007/s40304-023-00365-3 doi: 10.1007/s40304-023-00365-3
    [7] T. Zhou, L. Zhu, C. Xu, R. Li, Model-free forward screening via cumulative divergence, J. Am. Stat. Assoc., 115 (2020), 1393–1405. https://doi.org/10.1080/01621459.2019.1632078 doi: 10.1080/01621459.2019.1632078
    [8] L. Su, X. Zheng, A martingale-difference-divergence-based test for specification, Econ. Lett., 156 (2017), 162–167. https://doi.org/10.1016/j.econlet.2017.05.002 doi: 10.1016/j.econlet.2017.05.002
    [9] Z. Tian, T. Lai, Z. Zhang, Variation of conditional mean and its application in ultrahigh dimensional feature screening, Commun. Stat. Theory Methods, 54 (2024), 352–382. https://doi.org/10.1080/03610926.2024.2310690 doi: 10.1080/03610926.2024.2310690
    [10] T. Park, X. Shao, S. Yao, Partial martingale difference correlation, Electron. J. Stat., 9 (2015), 1492–1517. https://doi.org/10.1214/15-EJS1047 doi: 10.1214/15-EJS1047
    [11] G. J. Székely, M. L. Rizzo, Partial distance correlation with methods for dissimilarities, Ann. Stat., 42 (2014), 2382–2412. https://doi.org/10.1214/14-AOS1255 doi: 10.1214/14-AOS1255
    [12] L. Cai, X. Guo, W. Zhong, Test and measure for partial mean dependence based on machine learning methods, J. Am. Stat. Assoc., 120 (2025), 833–845. https://doi.org/10.1080/01621459.2024.2366030 doi: 10.1080/01621459.2024.2366030
    [13] A. R. Lundborg, I. Kim, R. D. Shah, R. J. Samworth, The projected covariance measure for assumption-lean variable significance testing, Ann. Stat., 52 (2024), 2851–2878. https://doi.org/10.1214/24-AOS2447 doi: 10.1214/24-AOS2447
    [14] B. D. Williamson, P. B. Gilbert, M. Carone, N. Simon, Nonparametric variable importance assessment using machine learning techniques, Biometrics, 77 (2021), 9–22. https://doi.org/10.1111/biom.13392 doi: 10.1111/biom.13392
    [15] B. Dai, X. Shen, W. Pan, Significance tests of feature relevance for a black-box learner, IEEE Trans. Neural Netw. Learn. Syst., 35 (2022), 1898–1911. https://doi.org/10.1109/TNNLS.2022.3185742 doi: 10.1109/TNNLS.2022.3185742
    [16] B. D. Williamson, P. B. Gilbert, N. R. Simon, M. Carone, A general framework for inference on algorithm-agnostic variable importance, J. Am. Stat. Assoc., 118 (2023), 1645–1658. https://doi.org/10.1080/01621459.2021.2003200 doi: 10.1080/01621459.2021.2003200
    [17] T. Lai, Z. Zhang, Y. Wang, L. Kong, Testing independence of functional variables by angle covariance, J. Multivar. Anal., 182 (2021), 104711. https://doi.org/10.1016/j.jmva.2020.104711 doi: 10.1016/j.jmva.2020.104711
    [18] Y. Zhou, Y. Zhang, L. Zhu, A projective approach to conditional independence test for dependent processes, J. Bus. Econ. Stat., 40 (2022), 398–407. https://doi.org/10.1080/07350015.2020.1826952 doi: 10.1080/07350015.2020.1826952
    [19] F. Ferraty, A. Mas, P. Vieu, Nonparametric regression on functional data: inference and practical aspects, Aust. N. Z. J. Stat., 49 (2007), 267–286. https://doi.org/10.1111/j.1467-842X.2007.00480.x doi: 10.1111/j.1467-842X.2007.00480.x
    [20] F. Ferraty, A. Laksaci, A. Tadj, P. Vieu, Kernel regression with functional response, Electron. J. Stat., 5 (2011), 159–171. https://doi.org/10.1214/11-EJS600 doi: 10.1214/11-EJS600
    [21] Z. Jin, D. S. Matteson, Generalizing distance covariance to measure and test multivariate mutual dependence via complete and incomplete V-statistics, J. Multivar. Anal., 168 (2018), 304–322. https://doi.org/10.1016/j.jmva.2018.08.006 doi: 10.1016/j.jmva.2018.08.006
    [22] F. Yao, H. G. Müller, J. L. Wang, Functional data analysis for sparse longitudinal data, J. Am. Stat. Assoc., 100 (2005), 577–590. https://doi.org/10.1198/016214504000001745 doi: 10.1198/016214504000001745
    [23] R. Ghosal, A. Maity, T. Clark, S. B. Longo, Variable selection in functional linear concurrent regression, J. R. Stat. Soc. Ser. C Appl. Stat., 69 (2020), 565–587. https://doi.org/10.1111/rssc.12408 doi: 10.1111/rssc.12408
    [24] N. Pfister, P. Bühlmann, B. Schölkopf, J. Peters, Kernel-based tests for joint independence, J. R. Stat. Soc. Ser. B Stat. Methodol., 80 (2017), 5–31. https://doi.org/10.1111/rssb.12235 doi: 10.1111/rssb.12235
  • Reader Comments
  • © 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Electronic Research Archive
1.1 1.7

Metrics

Article views(426) PDF downloads(18) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(3)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog