Hostname: page-component-78857b5c4d-vwtgk Total loading time: 0 Render date: 2023-08-04T00:52:20.634Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": false, "useRatesEcommerce": true } hasContentIssue false

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
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

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.

Type
Research Article
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): 108123.CrossRefGoogle 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): 207220.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): 177201.Google Scholar
4. Erdös, P. (1949). On a theorem of Hsu and Robbins. The Annals of Mathematical Statistics 20(2): 286291.CrossRefGoogle Scholar
5. Fan, Y. (1990). Consistent nonparametric multiple regression for dependent heterogeneous processes: The fixed design case. Journal of Multivariate Analysis 33: 7288.CrossRefGoogle 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. 141151.CrossRefGoogle 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: 1320.CrossRefGoogle Scholar
8. Gut, A. 1992. Complete convergence for arrays. Periodica Mathematica Hungarica 25(1): 5175.CrossRefGoogle 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): 918.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: 2531.Google Scholar
11. Joag-Dev, K. & Proschan, F. (1983). Negative association of random variables with applications. The Annals of Statistics 11(1): 286295.CrossRefGoogle Scholar
12. Katz, M.L. (1963). The probability in the tail of a distribution. The Annals of Mathematical Statistics 34(1): 312318.CrossRefGoogle 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.CrossRefGoogle Scholar
14. Kuczmaszewska, A. (2010). On complete convergence in Marcinkiewicz–Zygmund type SLLN for negatively associated random variables. Acta Mathematica Hungarica 128(1–2): 116130.CrossRefGoogle 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): 120.CrossRefGoogle 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: 227245.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): 419432.CrossRefGoogle Scholar
18. Roussas, G.G. (1989). Consistent regression estimation with fixed design points under dependence conditions. Statistics and Probability Letters 8: 4150.CrossRefGoogle 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: 262291.CrossRefGoogle 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): 343356.CrossRefGoogle Scholar
21. Shen, A.T., Zhang, Y., & Volodin, A. (2015). Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables. Metrika 78: 295311.CrossRefGoogle 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): 606621.Google 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): 251268.CrossRefGoogle Scholar
24. Stone, C.J. (1977). Consistent nonparametric regression. The Annals of Statistics 5: 595645.CrossRefGoogle Scholar
25. Sung, S.H. (2009). Moment inequalities and complete moment convergence. Journal of Inequalities and Applications, Article ID 271265, 14 pages.CrossRefGoogle Scholar
26. Tran, L., Roussas, G., Yakowitz, S., & Truong, Van B. (1996). Fixed-design regression for linear time series. The Annals of Statistics 24: 975991.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): 119132.CrossRefGoogle 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): 585596.CrossRefGoogle 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): 607629.CrossRefGoogle 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): 449468.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.CrossRefGoogle Scholar