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Multivariate Universal Local Linear Kernel Estimators in Nonparametric Regression: Uniform Consistency

Author

Listed:
  • Yuliana Linke

    (Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia
    Department of Probability and Mathematical Statistics, Novosibirsk State University, 630090 Novosibirsk, Russia)

  • Igor Borisov

    (Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia
    Department of Probability and Mathematical Statistics, Novosibirsk State University, 630090 Novosibirsk, Russia)

  • Pavel Ruzankin

    (Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia
    Department of Probability and Mathematical Statistics, Novosibirsk State University, 630090 Novosibirsk, Russia)

  • Vladimir Kutsenko

    (Department of Probability Theory, Lomonosov Moscow State University, 119234 Moscow, Russia
    Department of Epidemiology of Noncommunicable Diseases, National Medical Research Center for Therapy and Preventive Medicine, 101990 Moscow, Russia)

  • Elena Yarovaya

    (Department of Probability Theory, Lomonosov Moscow State University, 119234 Moscow, Russia
    Department of Epidemiology of Noncommunicable Diseases, National Medical Research Center for Therapy and Preventive Medicine, 101990 Moscow, Russia)

  • Svetlana Shalnova

    (Department of Epidemiology of Noncommunicable Diseases, National Medical Research Center for Therapy and Preventive Medicine, 101990 Moscow, Russia)

Abstract

In this paper, for a wide class of nonparametric regression models, new local linear kernel estimators are proposed that are uniformly consistent under close-to-minimal and visual conditions on design points. These estimators are universal in the sense that their designs can be either fixed and not necessarily satisfying the traditional regularity conditions, or random, while not necessarily consisting of independent or weakly dependent random variables. With regard to the design elements, only dense filling of the regression function domain with the design points without any specification of their correlation is assumed. This study extends the dense data methodology and main results of the authors’ previous work for the case of regression functions of several variables.

Suggested Citation

  • Yuliana Linke & Igor Borisov & Pavel Ruzankin & Vladimir Kutsenko & Elena Yarovaya & Svetlana Shalnova, 2024. "Multivariate Universal Local Linear Kernel Estimators in Nonparametric Regression: Uniform Consistency," Mathematics, MDPI, vol. 12(12), pages 1-23, June.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:12:p:1890-:d:1417300
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    References listed on IDEAS

    as
    1. Shen, Jia & Xie, Yuan, 2013. "Strong consistency of the internal estimator of nonparametric regression with dependent data," Statistics & Probability Letters, Elsevier, vol. 83(8), pages 1915-1925.
    2. Yuliana Linke & Igor Borisov, 2022. "Insensitivity of Nadaraya–Watson estimators to design correlation," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 51(19), pages 6909-6918, October.
    3. Oliver Linton & David Jacho-Chávez, 2010. "On internally corrected and symmetrized kernel estimators for nonparametric regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(1), pages 166-186, May.
    4. Ling Zhou & Huazhen Lin & Hua Liang, 2018. "Efficient Estimation of the Nonparametric Mean and Covariance Functions for Longitudinal and Sparse Functional Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(524), pages 1550-1564, October.
    5. Igor S. Borisov & Yuliana Yu. Linke & Pavel S. Ruzankin, 2021. "Universal weighted kernel-type estimators for some class of regression models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(2), pages 141-166, February.
    6. Tang, Xufei & Xi, Mengmei & Wu, Yi & Wang, Xuejun, 2018. "Asymptotic normality of a wavelet estimator for asymptotically negatively associated errors," Statistics & Probability Letters, Elsevier, vol. 140(C), pages 191-201.
    7. Linton, Oliver & Wang, Qiying, 2016. "Nonparametric Transformation Regression With Nonstationary Data," Econometric Theory, Cambridge University Press, vol. 32(1), pages 1-29, February.
    8. Hansen, Bruce E., 2008. "Uniform Convergence Rates For Kernel Estimation With Dependent Data," Econometric Theory, Cambridge University Press, vol. 24(3), pages 726-748, June.
    9. Roussas, George G., 1990. "Nonparametric regression estimation under mixing conditions," Stochastic Processes and their Applications, Elsevier, vol. 36(1), pages 107-116, October.
    10. Qiying Wang & Peter C. B. Phillips, 2009. "Structural Nonparametric Cointegrating Regression," Econometrica, Econometric Society, vol. 77(6), pages 1901-1948, November.
    11. Georgiev, Alexander A., 1990. "Nonparametric multiple function fitting," Statistics & Probability Letters, Elsevier, vol. 10(3), pages 203-211, August.
    12. Gao, Jiti & Kanaya, Shin & Li, Degui & Tjøstheim, Dag, 2015. "Uniform Consistency For Nonparametric Estimators In Null Recurrent Time Series," Econometric Theory, Cambridge University Press, vol. 31(5), pages 911-952, October.
    13. Gu, Wentao & Roussas, George G. & Tran, Lanh T., 2007. "On the convergence rate of fixed design regression estimators for negatively associated random variables," Statistics & Probability Letters, Elsevier, vol. 77(12), pages 1214-1224, July.
    14. Yao, Fang, 2007. "Asymptotic distributions of nonparametric regression estimators for longitudinal or functional data," Journal of Multivariate Analysis, Elsevier, vol. 98(1), pages 40-56, January.
    15. Liang, Han-Ying & Jing, Bing-Yi, 2005. "Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences," Journal of Multivariate Analysis, Elsevier, vol. 95(2), pages 227-245, August.
    16. Shuzhuan Zheng & Lijian Yang & Wolfgang K. Härdle, 2014. "A Smooth Simultaneous Confidence Corridor for the Mean of Sparse Functional Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(506), pages 661-673, June.
    17. Jingping Gu & Qi Li & Jui-Chung Yang, 2015. "Multivariate Local Polynomial Kernel Estimators: Leading Bias and Asymptotic Distribution," Econometric Reviews, Taylor & Francis Journals, vol. 34(6-10), pages 979-1010, December.
    18. Linke, Yu.Yu. & Borisov, I.S., 2017. "Constructing initial estimators in one-step estimation procedures of nonlinear regression," Statistics & Probability Letters, Elsevier, vol. 120(C), pages 87-94.
    19. Yuliana Linke, 2019. "Asymptotic properties of one-step M-estimators," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 48(16), pages 4096-4118, August.
    20. Yao, Fang & Muller, Hans-Georg & Wang, Jane-Ling, 2005. "Functional Data Analysis for Sparse Longitudinal Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 577-590, June.
    21. Wang, Qiying & Phillips, Peter C.B., 2009. "Asymptotic Theory For Local Time Density Estimation And Nonparametric Cointegrating Regression," Econometric Theory, Cambridge University Press, vol. 25(3), pages 710-738, June.
    22. Emmanuel Rio, 2009. "Moment Inequalities for Sums of Dependent Random Variables under Projective Conditions," Journal of Theoretical Probability, Springer, vol. 22(1), pages 146-163, March.
    23. Elias Masry, 1996. "Multivariate Local Polynomial Regression For Time Series:Uniform Strong Consistency And Rates," Journal of Time Series Analysis, Wiley Blackwell, vol. 17(6), pages 571-599, November.
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