Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF AANA RANDOM VARIABLES AND ITS APPLICATION IN NONPARAMETRIC REGRESSION MODELS

  • Shen, Aiting (School of Mathematical Sciences Anhui University) ;
  • Zhang, Yajing (School of Mathematical Sciences Anhui University)
  • Received : 2020.01.16
  • Accepted : 2020.07.21
  • Published : 2021.03.01
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Abstract

In this paper, we main study the strong law of large numbers and complete convergence for weighted sums of asymptotically almost negatively associated (AANA, in short) random variables, by using the Marcinkiewicz-Zygmund type moment inequality and Roenthal type moment inequality for AANA random variables. As an application, the complete consistency for the weighted linear estimator of nonparametric regression models based on AANA errors is obtained. Finally, some numerical simulations are carried out to verify the validity of our theoretical result.

Keywords

Acknowledgement

The authors are most grateful to the Editor and anonymous referee for carefully reading the manuscript and valuable suggestions which helped in improving an earlier version of this paper.

References

  1. J.-I. Baek, Almost sure convergence for asymptotically almost negatively associated random variable sequences, Commun. Korean Stat. Soc. 16 (2009), no. 6, 1013-1022.
  2. J.-I. Baek, I.-B. Choi, and S.-L. Niu, On the complete convergence of weighted sums for arrays of negatively associated variables, J. Korean Statist. Soc. 37 (2008), no. 1, 73-80. https://doi.org/10.1016/j.jkss.2007.08.001
  3. G.-H. Cai and B.-C. Guo, Complete convergence for weighted sums of sequences of AANA random variables, Glasg. Math. J. 50 (2008), no. 3, 351-357. https://doi.org/10.1017/S0017089508004254
  4. T. K. Chandra and S. Ghosal, The strong law of large numbers for weighted averages under dependence assumptions, J. Theoret. Probab. 9 (1996), no. 3, 797-809. https://doi.org/10.1007/BF02214087
  5. T. K. Chandra and S. Ghosal, Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables, Acta Math. Hungar. 71 (1996), no. 4, 327-336. https://doi.org/10.1007/BF00114421
  6. Y. Fan, Consistent nonparametric multiple regression for dependent heterogeneous processes: the fixed design case, J. Multivariate Anal. 33 (1990), no. 1, 72-88. https://doi.org/10.1016/0047-259X(90)90006-4
  7. A. A. Georgiev, Consistent nonparametric multiple regression: the fixed design case, J. Multivariate Anal. 25 (1988), no. 1, 100-110. https://doi.org/10.1016/0047-259X(88)90155-8
  8. A. A. Georgiev and W. Greblicki, Nonparametric function recovering from noisy observations, J. Statist. Plann. Inference 13 (1986), no. 1, 1-14. https://doi.org/10.1016/0378-3758(86)90114-X
  9. P. L. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 25-31. https://doi.org/10.1073/pnas.33.2.25
  10. H. Huang, H. Zou, and Y. Yi, Complete moment convergence for weighted sums of arrays of rowwise asymptotically almost negatively associated random variables, Adv. Math. (China) 48 (2019), no. 1, 110-120. https://doi.org/10.1007/jhep01(2019)051
  11. K. Joag-Dev and F. Proschan, Negative association of random variables, with applications, Ann. Statist. 11 (1983), no. 1, 286-295. https://doi.org/10.1214/aos/1176346079
  12. H.-G. Muller, Weak and universal consistency of moving weighted averages, Period. Math. Hungar. 18 (1987), no. 3, 241-250. https://doi.org/10.1007/BF01848087
  13. D. H. Qiu and X. Q. Yang, Strong laws of large numbers for weighted sums of identically distributed NA random variables, J. Math. Res. Exposition 26 (2006), no. 4, 778-784.
  14. G. G. Roussas, Consistent regression estimation with fixed design points under dependence conditions, Statist. Probab. Lett. 8 (1989), no. 1, 41-50. https://doi.org/10.1016/0167-7152(89)90081-3
  15. A. Shen, Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models, Abstr. Appl. Anal. 2013 (2013), Art. ID 862602, 9 pp. https://doi.org/10.1155/2013/862602
  16. A. Shen and D. Shi, Lr convergence for arrays of rowwise asymptotically almost negatively associated random variables, Comm. Statist. Theory Methods 46 (2017), no. 23, 11801-11812. https://doi.org/10.1080/03610926.2017.1285932
  17. A. Shen, R. Wu, Y. Chen, and Y. Zhou, Complete convergence of the maximum partial sums for arrays of rowwise of AANA random variables, Discrete Dyn. Nat. Soc. 2013 (2013), Art. ID 741901, 7 pp. https://doi.org/10.1155/2013/741901
  18. C. J. Stone, Consistent nonparametric regression, Ann. Statist. 5 (1977), no. 4, 595-645. https://doi.org/10.1214/aos/1176343886
  19. L. Tran, G. Roussas, S. Yakowitz, and B. Truong Van, Fixed-design regression for linear time series, Ann. Statist. 24 (1996), no. 3, 975-991. https://doi.org/10.1214/aos/ 1032526952
  20. X. Wang, X. Deng, L. Zheng, and S. Hu, Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications, Statistics 48 (2014), no. 4, 834-850. https://doi.org/10.1080/02331888.2013.800066
  21. X. Wang, S. Hu, and W. Yang, Convergence properties for asymptotically almost negatively associated sequence, Discrete Dyn. Nat. Soc. 2010 (2010), Art. ID 218380, 15 pp. https://doi.org/10.1155/2010/218380
  22. X. Wang, A. Shen, and X. Li, A note on complete convergence of weighted sums for array of rowwise AANA random variables, J. Inequal. Appl. 2013 (2013), 359, 13 pp. https://doi.org/10.1186/1029-242X-2013-359
  23. X. Wang, C. Xu, T. Hu, A. Volodin, and S. Hu, On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models, TEST 23 (2014), no. 3, 607-629. https://doi.org/10.1007/s11749-014-0365-7
  24. Q. Y. Wu, Limit theorems of probability theory for mixed sequences, Beijing, Science Press of China, 2006.
  25. Y. Wu, X. Wang, and S. Hu, Complete moment convergence for weighted sums of weakly dependent random variables and its application in nonparametric regression model, Statist. Probab. Lett. 127 (2017), 56-66. https://doi.org/10.1016/j.spl.2017.03.027
  26. M. Xi, X. Deng, X. Wang, and Z. Cheng, Lp convergence and complete convergence for weighted sums of AANA random variables, Comm. Statist. Theory Methods 47 (2018), no. 22, 5604-5613. https://doi.org/10.1080/03610926.2017.1397173
  27. D. Yuan and J. An, Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications, Sci. China Ser. A 52 (2009), no. 9, 1887-1904. https://doi.org/10.1007/s11425-009-0154-z
  28. N. Zhang and Y. Lan, Rosenthal's inequalities for asymptotically almost negatively associated random variables under upper expectations, Chin. Ann. Math. Ser. B 40 (2019), no. 1, 117-130. https://doi.org/10.1007/s11401-018-0122-4