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THE CONSISTENCY FOR THE WEIGHTED ESTIMATOR OF NON-PARAMETRIC REGRESSION MODEL BASED ON WIDELY ORTHANT-DEPENDENT ERRORS

Published online by Cambridge University Press:  03 July 2017

Hao Xia ,
Yi Wu ,
Xinran Tao  and
Xuejun Wang
Hao Xia
Affiliation:
School of Mathematical Sciences, Anhui University, Hefei 230601, People's Republic of China E-mail: 2943004220@qq.com; wuyi8702@163.com; 715231263@qq.com; wxjahdx2000@126.com
Yi Wu
Affiliation:
School of Mathematical Sciences, Anhui University, Hefei 230601, People's Republic of China E-mail: 2943004220@qq.com; wuyi8702@163.com; 715231263@qq.com; wxjahdx2000@126.com
Xinran Tao
Affiliation:
School of Mathematical Sciences, Anhui University, Hefei 230601, People's Republic of China E-mail: 2943004220@qq.com; wuyi8702@163.com; 715231263@qq.com; wxjahdx2000@126.com
Xuejun Wang
Affiliation:
School of Mathematical Sciences, Anhui University, Hefei 230601, People's Republic of China E-mail: 2943004220@qq.com; wuyi8702@163.com; 715231263@qq.com; wxjahdx2000@126.com
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Abstract

In this paper, the complete consistency for the weighted estimator of non-parametric regression model based on widely orthant-dependent errors is established, where the restriction imposed on the dominating coefficient g(n) is very general. Moreover, under some stronger moment condition, we further obtain the convergence rate of the complete consistency, where the assumption on the dominating coefficient g(n) is also very general.

Keywords

complete consistencyconvergence ratenon-parametric regression modelweighted estimatorwidely orthant-dependent errors
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 32 , Issue 3 , July 2018 , pp. 469 - 481
Copyright
Copyright © Cambridge University Press 2017 

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References

1.Fan, Y. (1990). Consistent nonparametric multiple regression for dependent heterogeneous processes: The fixed design case. Journal of Multivariate Analysis 33: 72–88.Google Scholar
2.Georgiev, A.A. (1985). Local properties of function fitting estimates with applications to system identification. In Grossmann, W. et al. (eds.), Mathematical statistics and applications, volume B, Proceedings 4th Pannonian Symposium on Mathematical Statistics, 4–10 September 1983. Austria, Reidel, Dordrecht: Bad Tatzmannsdorf, pp. 141–151.Google Scholar
3.Georgiev, A.A. (1988). Consistent nonparametric multiple regression: The fixed design case. Journal of Multivariate Analysis 25(1): 100–110.Google Scholar
4.He, W., Cheng, D.Y., & Wang, Y.B. (2013). Asymptotic lower bounds of precise large deviations with nonnegative and dependent random variables. Statistics and Probability Letters 83: 331–338.Google Scholar
5.Hu, S.H., Zhu, C.H., Chen, Y.B., & Wang, L.C. (2002). Fixed-design regression for linear time series. Acta Mathematica Scientia, Series B (English Edition) 22: 9–18.Google Scholar
6.Hu, S.H., Pan, G.M., & Gao, Q.B. (2003). Estimate problem of regression models with linear process errors. Applied Mathematics: A Journal of Chinese Universities 18A(1): 81–90.Google Scholar
7.Hsu, P.L. & Robbins, H. (1947). Complete convergence and the law of large numbers. Proceedings of the National Academy of Sciences of the United States of America 33: 25–31.Google Scholar
8.Hu, T.Z. (2000). Negatively superadditive dependence of random variables with applications. Chinese Journal of Applied Probability and Statistics 16: 133–144.Google Scholar
9.Joag-Dev, K. & Proschan, F. (1983). Negative association of random variables with applications. The Annals of Statistics 11: 286–295.Google Scholar
10.Lehmann, E.L. (1966). Some concepts of dependence. The Annals of Mathematical Statistics 37: 1137–1153.Google Scholar
11.Liang, H.Y. & Jing, B.Y. (2005). Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences. Journal of Multivariate Analysis 95: 227–245.Google Scholar
12.Liu, L. (2009). Precise large deviations for dependent random variables with heavy tails. Statistics and Probability Letters 79: 1290–1298.Google Scholar
13.Roussas, G.G. (1989). Consistent regression estimation with fixed design points under dependence conditions. Statistics and Probability Letters 8: 41–50.Google Scholar
14.Roussas, G.G., Tran, L.T., & Ioannides, D.A. (1992). Fixed design regression for time series: Asymptotic normality. Journal of Multivariate Analysis 40: 262–291.Google Scholar
15.Shen, A.T. (2013). Bernstein-type inequality for widely dependent sequence and its application to nonpara- metric regression models. Abstract and Applied Analysis 2013, 9 pages, Article ID 862602.Google Scholar
16.Shen, A.T. (2013). On the strong convergence rate for weighted sums of arrays of rowwise negatively orthant dependent random variables. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas (RACSAM) 107(2): 257–271.Google Scholar
17.Shen, A.T. (2014). On asymptotic approximation of inverse moments for a class of nonnegative random variables. Statistics: A Journal of Theoretical and Applied Statistics 48(6): 1371–1379.Google Scholar
18.Shen, A.T., Zhang, Y., & Volodin, A. (2015). Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables. Metrika 78: 295–311.Google Scholar
19.Stone, C.J. (1977). Consistent nonparametric regression. The Annals of Statistics 5: 595–645.Google Scholar
20.Tran, L., Roussas, G., Yakowitz, S., & Van Truong, B. (1996). Fixed-design regression for linear time series. The Annals of Statistics 24: 975–991.Google Scholar
21.Wang, K.Y., Wang, Y.B., & Gao, Q.W. (2013). Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate. Methodology and Computing in Applied Probability 15: 109–124.Google Scholar
22.Wang, X.J., Xu, C., Hu, T.-C., Volodin, A., & Hu, S.H. (2014). On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. TEST 23: 607–629.Google Scholar
23.Wang, X.J., Zheng, L.L., Xu, C., & Hu, S.H. (2015). Complete consistency for the estimator of nonparametric regression models based on extended negatively dependent errors. Statistics: A Journal of Theoretical and Applied Statistics 49: 396–407.Google Scholar
24.Wang, Y.B. & Cheng, D.Y. (2011). Basic renewal theorems for random walks with widely dependent increments. Journal of Mathematical Analysis and Applications 384: 597–606.Google Scholar
25.Wang, Y.B., Cui, Z.L., Wang, K.Y., & Ma, X.L. (2012). Uniform asymptotics of the finite-time ruin probability for all times. Journal of Mathematical Analysis and Applications 390: 208–223.Google Scholar
26.Wu, Q.Y. (2006). Probability limit theory for mixing sequences. Beijing: Science Press of China.Google Scholar
27.Yang, W.Z., Liu, T.T., Wang, X.J., & Hu, S.H. (2014). On the Bahadur representation of sample quantiles for widely orthant dependent sequences. Filomat 28: 1333–1343.Google Scholar
28.Yang, W.Z., Xu, H.Y., Chen, L., & Hu, S.H. (2016). Complete consistency of estimators for regression models based on extended negatively dependent errors. Statistical Papers, in press. DOI:10.1007/s00362-016-0771-x.Google Scholar