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Complete convergence for weighted sums of widely orthant-dependent random variables and its statistical application

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Abstract

In this paper, we investigate the complete convergence for weighted sums of widely orthant-dependent (WOD, for short) random variables. Our results extend the corresponding ones of Chen and Sung (Stat Probab Lett 154, 2019) to a much more general type of complete convergence. As an application of our main results, we establish the complete consistency for the estimator in the nonparametric regression models and provide a simulation study to assess the finite sample performance of the theoretical results.

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References

  1. Baum, L.E., Katz, M.: Convergence rates in the law of large numbers. Trans. Am. Math. Soc. 120, 108–123 (1965)

    Article  MathSciNet  Google Scholar 

  2. Chen, P.Y., Sung, S.H.: A Spitzer-type law of large numbers for widely orthant dependent random variables. Stat. Probab. Lett. 154, Article ID 108544 (2019)

  3. Chen, Z., Lu, C., Shen, Y., Wang, R., Wang, X.J.: On the complete and complete moment convergence for weighted sums of ANA random variables and applications. J. Stat. Comput. Simul. 89(15), 2871–2898 (2019)

    Article  MathSciNet  Google Scholar 

  4. Deng, X., Wang, X.J.: On complete convergence for extended independent random variables under sub-linear expectations. J. Korean Math. Soc. 57(3), 553–570 (2020)

    MathSciNet  MATH  Google Scholar 

  5. Engle, R., Granger, C., Rice, J., Weiss, A.: Nonparametric estimates of the relation between weather and electricity sales. J. Am. Stat. Assoc. 81, 310–320 (1986)

    Article  Google Scholar 

  6. Gao, M.M., Wang, K.Y., Chen, L.M.: Precise large deviations for widely orthant dependent random variables with different distributions. J. Inequal. Appl. 2018, Article ID 21 (2018)

  7. Georgiev, A.A.: Local properties of function fitting estimates with applications to system identification. In: Grossmann, W., et al. (eds.) Mathematical Statistics and Applications. Proceedings 4th Pannonian Symposium on Mathematical Statistics, 4–10, September 1983, vol. B, pp. 141–151. Bad Tatzmannsdorf, Austria, Reidel, Dordrecht (1985)

  8. He, Q.H.: Consistency of the Priestley-Chao estimator in nonparametric regression model with widely orthant dependent errors. J. Inequal. Appl. 2019, Article ID 64 (2019)

  9. Hosseini, S.M., Nezakati, A.: Complete moment convergence for the dependent linear processes with random coefficients. Acta Math. Sin. Engl. Ser. 35, 1321–1333 (2019)

    Article  MathSciNet  Google Scholar 

  10. Hsu, P.L., Robbins, H.: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33, 25–31 (1947)

    Article  MathSciNet  Google Scholar 

  11. Joag-Dev, K., Proschan, F.: Negative association of random variables with applications. Ann. Stat. 11(1), 286–295 (1983)

    Article  MathSciNet  Google Scholar 

  12. Li, P.H., Li, X.Q., Wu, K.H.: Complete convergence of randomly weighted END sequences and its application. J. Inequal. Appl. 2017, Article ID 182 (2017)

  13. Liang, Z.W., Wu, Q.Y.: Theorems of complete convergence and complete moment convergence for END random variables under sub-linear expectations. J. Inequal. Appl. 2019, Article ID 114 (2019)

  14. Lu, C., Chen, Z., Wang, X.J.: Complete \(f\)-moment convergence for widely orthant dependent random variables and its application in nonparametric models. Acta Math. Sin. Engl. Ser. 35(12), 1917–1936 (2019)

    Article  MathSciNet  Google Scholar 

  15. Lu, C., Li, X.Q., Wang, R., Wang, X.J.: Complete and complete moment convergence for randomly weighted sums of \(\rho ^{*}\)-mixing random variables and its applications. Statistics 54(1), 205–237 (2020)

    Article  MathSciNet  Google Scholar 

  16. Lu, D.W., Wang, L.N,: Complete moment convergence for the widely orthant dependent linear processes with random coefficients. Commun. Stat. Theory Methods (2020). https://doi.org/10.1080/03610926.2020.1756328

  17. Ning, M.M., Wu, C.Q., Shen, A.T.: Complete convergence of weighted sums of WNOD random variables and its applications. Stoch. Int. J. Probab. Stoch Process. 92(1), 24–45 (2020)

    Article  MathSciNet  Google Scholar 

  18. Shen, A.T., Li, X., Ning, M.M.: Strong convergence properties for weighted sums of WNOD random variables and its applications in nonparametric regression models. Stoch. Int. J. Probab. Stoch Process. (2020). https://doi.org/10.1080/17442508.2020.1734005

  19. Shen, A.T., Tao, H.L., Wang, X.J.: The asymptotic properties for the estimators of the survival function and failure rate function based on WOD samples. Stat. Pap. (2019). https://doi.org/10.1007/s00362-018-01064-w

  20. Shen, A.T., Zhang, Y., Volodin, A.: Applications of the Rosenthal-type inequality for negatively super-additive dependent random variables. Metrika 78, 295–311 (2015)

    Article  MathSciNet  Google Scholar 

  21. Spitzer, F.: A combinatorial lemma and its application to probability theory. Trans. Am. Math. Soc. 82, 323–339 (1956)

    Article  MathSciNet  Google Scholar 

  22. Stone, C.J.: Consistent nonparametric regression. Ann. Stat. 5, 595–620 (1977)

    Article  MathSciNet  Google Scholar 

  23. Sung, S.H.: Complete convergence for weighted sums of \(\rho ^*\)-mixing random variables. Discrete Dyn. Nat. Soc. 2010, Article ID 630608 (2010)

  24. Stout, W.F.: Almost Sure Convergence. Academic Press, New York (1974)

    MATH  Google Scholar 

  25. Wang, K.Y., Wang, Y.B., Gao, Q.W.: Uniform asymptotic for the finite-time ruin probability of dependent risk model with a constant interest rate. Methodol. Comput. Appl. Probab. 15(1), 109–124 (2013)

    Article  MathSciNet  Google Scholar 

  26. Wang, K.Y., Yang, Y., Lin, J.G.: Precise large deviations for widely orthant dependent random variables with dominatedly varying tails. Front. Math. China 7, 919–932 (2012)

    Article  MathSciNet  Google Scholar 

  27. Wang, X.J., Shen, A.T., Chen, Z.Y., Hu, S.H.: Complete convergence for weighted sums of NSD random variables and its application in the EV regression model. TEST 24(1), 166–184 (2015)

    Article  MathSciNet  Google Scholar 

  28. Wang, X.J., Xu, C., Hu, T.C., Volodin, A., Hu, S.H.: On the complete convergence for widely orthant-dependent random variables and its application in nonparametric regression models. TEST 23(3), 607–629 (2014)

    Article  MathSciNet  Google Scholar 

  29. Wang, X.J., Zheng, L.L., Xu, C., Hu, S.H.: Complete consistency for the estimator of nonparametric regression models based on extended negatively dependent errors. Statistics 49(2), 396–407 (2015)

    Article  MathSciNet  Google Scholar 

  30. Wu, Q.Y.: Probability Limit Theory for Mixing Sequences. Science Press of China, Beijing (2006)

    Google Scholar 

  31. Wu, Y., Wang, X.J., Hu, S.H.: Complete moment convergence for weighted sums of weakly dependent random variables and its application in nonparametric regression model. Stat. Probab. Lett. 127, 56–66 (2017)

    Article  MathSciNet  Google Scholar 

  32. Wu, Y., Wang, X.J., Hu, T.C., Cabrera, M.O., Volodin, A.: Limiting behavior for arrays of rowwise widely orthant dependent random variables under conditions of \(R\)–\(h\)-integrability and its applications. Stoch. Int. J. Probab. Stoch. Process. 91(6), 916–944 (2019)

    Article  Google Scholar 

  33. Wu, Y., Wang, X.J., Sung, S.H.: Complete moment convergence for array of rowwise negatively associated random variables and its application in non-parametric regression model. Probab. Eng. Inf. Sci. 32, 27–57 (2018)

    MathSciNet  MATH  Google Scholar 

  34. Wu, Y.F., Zhai, M.Q., Peng, J.Y.: On the complete convergence for weighted sums of extended negatively dependent random variables. J. Math. Inequal. 13(1), 251–260 (2019)

    MathSciNet  MATH  Google Scholar 

  35. Yan, J.G.: On complete convergence in Marcinkiewicz–Zygmund type SLLN for END random variables and its applications. Commun. Stat. Theory Methods 48(20), 5074–5098 (2019)

    Article  MathSciNet  Google Scholar 

  36. Yang, W.Z., Liu, T.T., Wang, X.J., Hu, S.H.: On the Bahadur representation of sample quantiles for widely orthant dependent sequence. Filomat 28(7), 1333–1343 (2014)

    Article  MathSciNet  Google Scholar 

  37. Zhang, R., Wu, Y., Xu, W., Wang, X.J.: On complete consistency for the weighted estimator of nonparametric regression models. RACSAM 113(3), 2319–2333 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. School of Mathematical Sciences, Anhui University, Hefei, 230601, People’s Republic of China

    Junjun Lang, Tianyun He, Lu Cheng, Chao Lu & Xuejun Wang

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  1. Junjun Lang
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  3. Lu Cheng
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Correspondence to Xuejun Wang.

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Supported by the National Natural Science Foundation of China (11671012, 11871072, 11701004, 11701005), the Natural Science Foundation of Anhui Province (1808085QA03, 1908085QA01, 1908085QA07), the Provincial Natural Science Research Project of Anhui Colleges (KJ2019A0001, KJ2019A0003) and the Students Innovative Training Project of Anhui University (S201910357342).

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Lang, J., He, T., Cheng, L. et al. Complete convergence for weighted sums of widely orthant-dependent random variables and its statistical application. Rev Mat Complut 34, 853–881 (2021). https://doi.org/10.1007/s13163-020-00369-5

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