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Consistent nonparametric multiple regression: The fixed design case

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  • Georgiev, Alexander A.

Abstract

Consider the nonparametric regression model Yi(n) = g(xi(n)) + [var epsilon]i(n), i = 1, ..., n, where g is an unknown function, the design points xi(n) are known and nonrandom, and [var epsilon]i(n)'s are independent random variables. The regressor is assumed to take values in A [subset of] Rp, and the regressand to be real valued. This paper studies the behavior of the general nonparametric estimate , where the weight function wni is of the form wni(x) = wni(x; xi(n), ..., xn(n)). Under suitable conditions, it is shown that the general linear smoother gn for the unknown regression function g is asymptotically pointwise unbiased, weak, mean square and complete consistent, and asymptotically normal. The results of the limit theorems can be applied to extend or improve the conditions of the estimates with various particular weights wni including all those known in the literature.

Suggested Citation

  • Georgiev, Alexander A., 1988. "Consistent nonparametric multiple regression: The fixed design case," Journal of Multivariate Analysis, Elsevier, vol. 25(1), pages 100-110, April.
    • Handle: RePEc:eee:jmvana:v:25:y:1988:i:1:p:100-110
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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. Aiting Shen & Siyao Zhang, 2021. "On Complete Consistency for the Estimator of Nonparametric Regression Model Based on Asymptotically Almost Negatively Associated Errors," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1285-1307, December.
    3. Bianco, Ana M. & Boente, Graciela & Sombielle, Susana, 2011. "Robust estimation for nonparametric generalized regression," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1986-1994.
    4. Li, Yongming & Wei, Chengdong & Xing, Guodong, 2011. "Berry-Esseen bounds for wavelet estimator in a regression model with linear process errors," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 103-110, January.
    5. Xuejun Wang & Yi Wu & Shuhe Hu & Nengxiang Ling, 2020. "Complete moment convergence for negatively orthant dependent random variables and its applications in statistical models," Statistical Papers, Springer, vol. 61(3), pages 1147-1180, June.
    6. 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.
    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. Thanh, Le Van & Yin, G., 2015. "Weighted sums of strongly mixing random variables with an application to nonparametric regression," Statistics & Probability Letters, Elsevier, vol. 105(C), pages 195-202.
    9. 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.
    10. Wenzhi Yang & Haiyun Xu & Ling Chen & Shuhe Hu, 2018. "Complete consistency of estimators for regression models based on extended negatively dependent errors," Statistical Papers, Springer, vol. 59(2), pages 449-465, June.
    11. Xuejun Wang & Yi Wu & Rui Wang & Shuhe Hu, 2021. "On consistency of wavelet estimator in nonparametric regression models," Statistical Papers, Springer, vol. 62(2), pages 935-962, April.
    12. Han-Ying Liang & Ya-Mei Liu, 2011. "Asymptotic normality of variance estimator in a heteroscedastic model with dependent errors," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(2), pages 351-365.
    13. Yi Wu & Xuejun Wang & Shuhe Hu & Lianqiang Yang, 2018. "Weighted version of strong law of large numbers for a class of random variables and its applications," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(2), pages 379-406, June.
    14. J. Fernández & W. Manteiga, 2001. "Generalized minimum distance estimators of a linear model with correlated errors," Statistical Papers, Springer, vol. 42(3), pages 353-373, July.
    15. Wentao Gu & Lanh Tran, 2009. "Fixed design regression for negatively associated random fields," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(3), pages 345-363.
    16. Yan, Ji Gao, 2018. "On Complete Convergence in Marcinkiewicz-Zygmund Type SLLN for END Random Variables and its Applications," IRTG 1792 Discussion Papers 2018-042, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".

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