Hostname: page-component-84b7d79bbc-fnpn6 Total loading time: 0 Render date: 2024-07-28T14:25:59.039Z Has data issue: false hasContentIssue false
  • English
  • Français

COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF ROWWISE NEGATIVELY ASSOCIATED RANDOM VARIABLES AND ITS APPLICATION IN NON-PARAMETRIC REGRESSION MODEL

Published online by Cambridge University Press:  16 January 2017

Yi Wu ,
Xuejun Wang  and
Soo Hak Sung
Yi Wu
Affiliation:
School of Mathematical Sciences, Anhui University, Hefei 230601, People's Republic of China E-mail: wxjahdx@126.com
Xuejun Wang
Affiliation:
School of Mathematical Sciences, Anhui University, Hefei 230601, People's Republic of China E-mail: wxjahdx@126.com
Soo Hak Sung
Affiliation:
Department of Applied Mathematics, Pai Chai University, Taejon 302-735, South Korea
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, some results on the complete moment convergence for arrays of rowwise negatively associated (NA, for short) random variables are established. The results obtained in this paper correct the corresponding one obtained in Ko [13] and also improve and generalize the corresponding ones of Kuczmaszewska [14] and Ko [13]. As an application of the main results, we present a result on complete consistency for the estimator in a non-parametric regression model based on NA errors. Finally, we provide a numerical simulation to verify the validity of our result.

Keywords

complete consistencycomplete convergencecomplete moment convergencenegatively associated random variablesnon-parametric regression model
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 32 , Issue 1 , January 2018 , pp. 37 - 57
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Baum, L.E. & Katz, M. (1965). Convergence rates in the law of large numbers. Transactions of the American Mathematical Society 120(1): 108–123.Google Scholar
2. Chow, Y.S. (1973). Delayed sums and Borel summability of independent, identically distributed random variables. Bulletin of the Institute of Mathematics, Academia Sinica 1(2): 207–220.Google Scholar
3. Chow, Y.S. (1988). On the rate of moment convergence of sample sums and extremes. Bulletin of the Institute of Mathematics, Academia Sinica 16(3): 177–201.Google Scholar
4. Erdös, P. (1949). On a theorem of Hsu and Robbins. The Annals of Mathematical Statistics 20(2): 286–291.CrossRefGoogle Scholar
5. Fan, Y. (1990). Consistent nonparametric multiple regression for dependent heterogeneous processes: The fixed design case. Journal of Multivariate Analysis 33: 72–88.Google Scholar
6. 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, Vol. B, Proceedings 4th Pannonian Symposium on Mathematical Statistics, 4–10 September 1983, Austria: Bad Tatzmannsdorf, Dordrecht: Reidel, pp. 141–151.Google Scholar
7. Guo, M.L. & Zhu, D.J. (2013). Equivalent conditions of complete moment convergence of weighted sums for ρ*-mixing sequence of random variables. Statistics and Probability Letters 83: 13–20.CrossRefGoogle Scholar
8. Gut, A. 1992. Complete convergence for arrays. Periodica Mathematica Hungarica 25(1): 51–75.Google Scholar
9. 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 22(1): 9–18.Google Scholar
10. Hsu, P.L. & Robbins, H. (1947). Complete convergence and the law of large numbers. Proceedings of the National Academy of Sciences USA 33: 25–31.Google Scholar
11. Joag-Dev, K. & Proschan, F. (1983). Negative association of random variables with applications. The Annals of Statistics 11(1): 286–295.Google Scholar
12. Katz, M.L. (1963). The probability in the tail of a distribution. The Annals of Mathematical Statistics 34(1): 312–318.Google Scholar
13. Ko, M.H. (2016). On complete moment convergence for nonstationary negatively associated random variables. Journal of Inequalities and Applications, Article ID 131, 12 pages.Google Scholar
14. Kuczmaszewska, A. (2010). On complete convergence in Marcinkiewicz–Zygmund type SLLN for negatively associated random variables. Acta Mathematica Hungarica 128(1–2): 116–130.Google Scholar
15. Liang, H.Y. & Jong-Il, Baek, 2016. Asymptotic normality of conditional density estimation with left-truncated and dependent data. Statistical Papers 57(1): 1–20.Google Scholar
16. 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.CrossRefGoogle Scholar
17. Liang, H.Y., Li, D.L., & Rosalsky, A. (2010). Complete moment and integral convergence for sums of negatively associated random variables. Acta Mathematica Sinica (English Series) 26(3): 419–432.Google Scholar
18. Roussas, G.G. (1989). Consistent regression estimation with fixed design points under dependence conditions. Statistics and Probability Letters 8: 41–50.Google Scholar
19. 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
20. Shao, Q.M. (2000). A comparison theorem on moment inequalities between negatively associated and independent random variables. Journal of Theoretical Probability 13(2): 343–356.Google Scholar
21. 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
22. Shen, A.T., Xue, M.X., & Volodin, A. (2016). Complete moment convergence for arrays of rowwise NSD random variables. Stochastics: An International Journal of Probability and Stochastic Processes 88(4): 606–621.CrossRefGoogle Scholar
23. Shen, A.T., Yao, M., Wang, W.J., & Volodin, A. (2016). Exponential probability inequalities for WNOD random variables and their applications. RACSAM 110(1): 251–268.Google Scholar
24. Stone, C.J. (1977). Consistent nonparametric regression. The Annals of Statistics 5: 595–645.Google Scholar
25. Sung, S.H. (2009). Moment inequalities and complete moment convergence. Journal of Inequalities and Applications, Article ID 271265, 14 pages.Google Scholar
26. Tran, L., Roussas, G., Yakowitz, S., & Truong, Van B. (1996). Fixed-design regression for linear time series. The Annals of Statistics 24: 975–991.Google Scholar
27. Wang, X.J. & Hu, S.H. (2014). Complete convergence and complete moment convergence for martingale difference sequence. Acta Mathematica Sinica (English Series) 30(1): 119–132.Google Scholar
28. Wang, X.J. & Si, Z.Y. (2015). Complete consistency of the estimator of nonparametric regression model under ND sequence. Statistical Papers 56(3): 585–596.Google Scholar
29. Wang, X.J., Xu, C., Hu, T.C., & Hu, S.H. (2014). On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. TEST 23(3): 607–629.Google Scholar
30. Wu, Y.F., Cabrea, M.O., & Volodin, A. (2014). Complete convergence and complete moment convergence for arrays of rowwise END random variables. Glasnik Matematički 49(69): 449–468.Google Scholar
31. 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, doi:10.1007/s00362-016-0771-x, in press.Google Scholar