Bulletin of the Korean Mathematical Society (대한수학회보)

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ALMOST SURE AND COMPLETE CONSISTENCY OF THE ESTIMATOR IN NONPARAMETRIC REGRESSION MODEL FOR NEGATIVELY ORTHANT DEPENDENT RANDOM VARIABLES

  • Ding, Liwang (School of Science Nanjing University of Science and Technology)
  • Received : 2018.12.28
  • Accepted : 2019.11.06
  • Published : 2020.01.31
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Abstract

In this paper, the author considers the nonparametric regression model with negatively orthant dependent random variables. The wavelet procedures are developed to estimate the regression function. For the wavelet estimator of unknown function g(·), the almost sure consistency is derived and the complete consistency is established under the mild conditions. Our results generalize and improve some known ones for independent random variables and dependent random variables.

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References

  1. A. Antoniadis, G. Gregoire, and I. W. McKeague, Wavelet methods for curve estimation, J. Amer. Statist. Assoc. 89 (1994), no. 428, 1340-1353. https://doi.org/10.1080/01621459.1994.10476873
  2. X. Bao, J. Lin, X. Wang, and Y. Wu, On complete convergence for weighted sums of arrays of rowwise END random variables and its statistical applications, Math. Slovaca 69 (2019), no. 1, 223-232. https://doi.org/10.1515/ms-2017-0216
  3. L. Ding, P. Chen, and Y. Li, Consistency for wavelet estimator in nonparametric regression model with extended negatively dependent samples, Statist. Papers (2018). https://doi.org/10.1007/s00362-018-1050-9
  4. L. Ding, Berry-Esseen bound of wavelet estimators in heteroscedastic regression model with random errors, Int. J. Comput. Math. 96 (2019), no. 4, 821-852. https://doi.org/10.1080/00207160.2018.1487958
  5. A. A. Georgiev, Local properties of function fitting estimates with application to system identification, in Mathematical statistics and applications, Vol. B (Bad Tatzmannsdorf, 1983), 141-151, Reidel, Dordrecht, 1985.
  6. 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
  7. 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
  8. P. Hall and P. Patil, Formulae for mean integrated squared error of nonlinear waveletbased density estimators, Ann. Statist. 23 (1995), no. 3, 905-928. https://doi.org/10.1214/aos/1176324628
  9. 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
  10. Y. Li, C. Wei, and G. Xing, Berry-Esseen bounds for wavelet estimator in a regression model with linear process errors, Statist. Probab. Lett. 81 (2011), no. 1, 103-110. https://doi.org/10.1016/j.spl.2010.09.024
  11. X. Li, W. Z. Yang, S. H. Hu, and X. J. Wang, The Bahadur representation for sample quantile under NOD sequence, J. Nonparametr. Stat. 23 (2011), no. 1, 59-65. https://doi.org/10.1080/10485252.2010.486033
  12. H.-Y. Liang, Asymptotic normality of wavelet estimator in heteroscedastic model with ${\alpha}$-mixing errors, J. Syst. Sci. Complex. 24 (2011), no. 4, 725-737. https://doi.org/10.1007/s11424-010-8354-8
  13. H.-Y. Liang and B.-Y. Jing, Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences, J. Multivariate Anal. 95 (2005), no. 2, 227-245. https://doi.org/10.1016/j.jmva.2004.06.004
  14. 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
  15. A. Shen, On the strong convergence rate for weighted sums of arrays of rowwise negatively orthant dependent random variables, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 107 (2013), no. 2, 257-271. https://doi.org/10.1007/s13398-012-0067-5
  16. A. Shen, Y. Zhang, and A. Volodin, Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables, Metrika 78 (2015), no. 3, 295-311. https://doi.org/10.1007/s00184-014-0503-y
  17. R. L. Taylor, R. F. Patterson, and A. Bozorgnia, A strong law of large numbers for arrays of rowwise negatively dependent random variables, Stochastic Anal. Appl. 20 (2002), no. 3, 643-656. https://doi.org/10.1081/SAP-120004118
  18. X. Wang and S. Hu, On consistency of least square estimators in the simple linear EV model with negatively orthant dependent errors, Electron. J. Stat. 11 (2017), no. 1, 1434-1463. https://doi.org/10.1214/17-EJS1263
  19. X. Wang, S. Hu, A. Shen, and W. Yang, An exponential inequality for a NOD sequence and a strong law of large numbers, Appl. Math. Lett. 24 (2011), no. 2, 219-223. https://doi.org/10.1016/j.aml.2010.09.007
  20. X. Wang, S. Hu, and A. I. Volodin, Strong limit theorems for weighted sums of NOD sequence and exponential inequalities, Bull. Korean Math. Soc. 48 (2011), no. 5, 923-938. https://doi.org/10.4134/BKMS.2011.48.5.923
  21. X. Wang, S. Hu, and W. Yang, Complete convergence for arrays of rowwise negatively orthant dependent random variables, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 106 (2012), no. 2, 235-245. https://doi.org/10.1007/s13398-011-0048-0
  22. X. Wang, S. Hu, W. Yang, and N. Ling, Exponential inequalities and inverse moment for NOD sequence, Statist. Probab. Lett. 80 (2010), no. 5-6, 452-461. https://doi.org/10.1016/j.spl.2009.11.023
  23. X. Wang and Z. Si, Complete consistency of the estimator of nonparametric regression model under ND sequence, Statist. Papers 56 (2015), no. 3, 585-596. https://doi.org/10.1007/s00362-014-0598-2
  24. X. Wang, Y. Wu, S. Hu, and N. Ling, Complete moment convergence for negatively orthant dependent random variables and its applications in statistical models, Statist. Papers 2018 (2018). https://doi.prg/10.1007/s00362-018-0983-3
  25. Q. Y. Wu, Probability Limit Theory for Mixing and Dependent Sequences, Science Press of China, Beijing, 2006
  26. Q. Y. Wu, Complete convergence for weighted sums of sequences of negatively dependent random variables, J. Probab. Stat. 2011 (2011), Art. ID 202015, 16 pp. https://doi.org/10.1155/2011/202015
  27. L. G. Xue and Q. Liu, Bootstrap approximation of wavelet estimates in a semiparameter regression model, Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 4, 763-778. https://doi.org/10.1007/s10114-010-7236-2
  28. S. C. Yang, Maximal moment inequality for partial sums of strong mixing sequences and application, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 6, 1013-1024. https://doi.org/10.1007/s10114-005-0841-9
  29. R. Zhang, Y. Wu, W. F. Xu, and X. J. Wang, On complete consistency for the weighted estimator of nonparametric regression models, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113 (2019), no. 3, 2319-2333. https://doi.org/10.1007/s13398-018-00621-0
  30. X. Zhou, Y. Xu, and J. Lin, Wavelet estimation in varying coefficient models for cen- sored dependent data, Statist. Probab. Lett. 122 (2017), 179-189. https://doi.org/10.1016/j.spl.2016.11.009