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Consistent nonparametric multiple regression for dependent heterogeneous processes: The fixed design case

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  • Fan, Y.

Abstract

Consider the nonparametric regression model Yi(n) = g(xi(n)) + [var epsilon]i(n), i = 1, ..., n, where g is an unknown regression function and assumed to be bounded and real valued on A [subset of] Rp, xi(n)'s are known and fixed design points and [var epsilon]i(n)'s are assumed to be both dependent and non-identically distributed random variables. This paper investigates the asymptotic properties of the general nonparametric regression estimator gn(x) = [Sigma]i = 1n Wni(x) Yi(n), where the weight function Wni(x) is of the form Wni(x) = Wni(x; x1(n), x2(n), ..., xn(n). The estimator gn(x) is shown to be weak, mean square error, and universal consistent under very general conditions on the temporal dependence and heterogeneity of [var epsilon]i(n)'s. Asymptotic distribution of the estimator is also considered.

Suggested Citation

  • Fan, Y., 1990. "Consistent nonparametric multiple regression for dependent heterogeneous processes: The fixed design case," Journal of Multivariate Analysis, Elsevier, vol. 33(1), pages 72-88, April.
  • Handle: RePEc:eee:jmvana:v:33:y:1990:i:1:p:72-88
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    Citations

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    Cited by:

    1. Aiting Shen & Ying Zhang & Andrei Volodin, 2015. "Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(3), pages 295-311, April.
    2. repec:eee:stapro:v:127:y:2017:i:c:p:56-66 is not listed on IDEAS
    3. Xuejun Wang & Yi Wu & Shuhe Hu, 2016. "Exponential probability inequality for $$m$$ m -END random variables and its applications," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(2), pages 127-147, February.
    4. Zhiyong Chen & Haibin Wang & Xuejun Wang, 2016. "The consistency for the estimator of nonparametric regression model based on martingale difference errors," Statistical Papers, Springer, vol. 57(2), pages 451-469, April.
    5. Xuejun Wang & Chen Xu & Tien-Chung Hu & Andrei Volodin & Shuhe Hu, 2014. "On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(3), pages 607-629, September.
    6. repec:spr:metrik:v:80:y:2017:i:6:d:10.1007_s00184-017-0618-z is not listed on IDEAS
    7. Yang, Shanchao, 2003. "Uniformly asymptotic normality of the regression weighted estimator for negatively associated samples," Statistics & Probability Letters, Elsevier, vol. 62(2), pages 101-110, April.
    8. Xuejun Wang & Zeyu Si, 2015. "Complete consistency of the estimator of nonparametric regression model under ND sequence," Statistical Papers, Springer, vol. 56(3), pages 585-596, August.
    9. Kapetanios, George, 2007. "Estimating deterministically time-varying variances in regression models," Economics Letters, Elsevier, vol. 97(2), pages 97-104, November.
    10. 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.
    11. Zhou, Xing-cai & Lin, Jin-guan, 2012. "A wavelet estimator in a nonparametric regression model with repeated measurements under martingale difference error’s structure," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1914-1922.

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