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Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables

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Abstract

In this paper, we give some applications of the Rosenthal-type inequality for a sequence of negatively superadditive dependent (NSD) random variables, which includes sequences of negatively associated random variables as a special case. The complete consistency for an estimator of a nonparametric regression model based on NSD errors is investigated. In addition, we extend Feller’s weak law of large numbers for independent and identically distributed random variables to the case of NSD random variables by using the Rosenthal-type inequality.

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References

  • Christofides TC, Vaggelatou E (2004) A connection between supermodular ordering and positive/negative association. J Multivar Anal 88:138–151

    Article  MATH  MathSciNet  Google Scholar 

  • Eghbal N, Amini M, Bozorgnia A (2010) Some maximal inequalities for quadratic forms of negative superadditive dependent random variables. Stat Probab Lett 80:587–591

    Article  MATH  MathSciNet  Google Scholar 

  • Eghbal N, Amini M, Bozorgnia A (2011) On the Kolmogorov inequalities for quadratic forms of dependent uniformly bounded random variables. Stat Probab Lett 81:1112–1120

    Article  MATH  MathSciNet  Google Scholar 

  • Fan Y (1990) Consistent nonparametric multiple regression for dependent heterogeneous processes. J Multivar Anal 33:72–88

    Article  MATH  Google Scholar 

  • Feller W (1946) A limit theorem for random variables with infinite moments. Am J Math 68:257–262

    Article  MATH  MathSciNet  Google Scholar 

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

  • Georgiev AA, Greblicki W (1986) Nonparametric function recovering from noisy observations. J Stat Plan Inference 13:1–14

    Article  MATH  MathSciNet  Google Scholar 

  • Georgiev AA (1988) Consistent nonparametric multiple regression: the fixed design case. J Multivar Anal 25:100–110

    Article  MATH  MathSciNet  Google Scholar 

  • Hu SH, Zhu CH, Chen YB, Wang LC (2002) Fixed-design regression for linear time series. Acta Mathematica Scientia 22B:9–18

    MathSciNet  Google Scholar 

  • Hu TZ (2000) Negatively superadditive dependence of random variables with applications. Chin J Appl Probab Stat 16:133–144

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Kemperman JHB (1977) On the FKG-inequalities for measures on a partially ordered space. Nederl Akad Wetensch Proc Ser A 80:313–331

    Article  MATH  MathSciNet  Google Scholar 

  • Liang HY, Jing BY (2005) Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences. J Multivar Anal 95:227–245

    Article  MATH  MathSciNet  Google Scholar 

  • Müller HG (1987) Weak and universal consistency of moving weighted averages. Periodica Mathematica Hungarica 18:241–250

    Article  MATH  MathSciNet  Google Scholar 

  • Roussas GG (1989) Consistent regression estimation with fixed design points under dependence conditions. Stat Probab Lett 8:41–50

    Article  MATH  MathSciNet  Google Scholar 

  • Roussas GG, Tran LT, Ioannides DA (1992) Fixed design regression for time series: asymptotic normality. J Multivar Anal 40:262–291

    Article  MATH  MathSciNet  Google Scholar 

  • Shen Y, Wang XJ, Yang WZ, Hu SH (2013a) Almost sure convergence theorem and strong stability for weighted sums of NSD random variables. Acta Math Sin Engl Ser 29(4):743–756

    Article  MATH  MathSciNet  Google Scholar 

  • Shen Y, Wang XJ, Hu SH (2013b) On the strong convergence and some inequalities for negatively superadditive dependent sequences. J Inequal Appl 2013:629

    Google Scholar 

  • Stone CJ (1977) Consistent nonparametric regression. Ann Stat 5:595–620

    Article  MATH  Google Scholar 

  • Tang XF (2013a) Some strong laws of large numbers for weighted sums of asymptotically almost negatively associated random variables. J Inequal Appl 2013:4

    Article  Google Scholar 

  • Tang XF (2013b) Strong convergence results for arrays of rowwise pairwise NQD random variables. J Inequal Appl 2013:102

  • Tran LT, Roussas GG, Yakowitz S, Van BT (1996) Fixed-design regression for linear time series. Ann Stat 24:975–991

    Article  MATH  Google Scholar 

  • Wang XJ, Li XQ, Hu SH, Yang WZ (2011) Strong limit theorems for weighted sums of negatively associated random variables. Stoch Anal Appl 29(1):1–14

    Article  MATH  MathSciNet  Google Scholar 

  • Wang XJ, Xia FX, Ge MM, Hu SH, Yang WZ (2012) Complete consistency of the estimator of nonparametric regression models based on \(\tilde{\rho }\)-mixing sequences. Abstract and Applied Analysis, Volume 2012, Article ID 907286, 12 p

  • Wang XJ, Deng X, Zheng LL, Hu SH (2013a) Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications. Stat J Theor Appl Stat. doi:10.1080/02331888.2013.800066

  • Wang XJ, Ge MM, Hu SH, Wang XZ (2013b) The strong consistency of the estimator of fixed-design regression model under negatively dependent sequences. Abstract and Applied Analysis, Volume 2013, Article ID 521618, 7 p

  • Wu QY, Jiang YY (2010a) Chover’s law of the iterated logarithm for negatively associated sequences. J Syst Sci Complex 23:293–302

  • Wu QY, Jiang YY (2010b) A law of the iterated logarithm of partial sums for NA random variables. J Korean Stat Soc 39:199–206

  • Wu QY (2010) A strong limit theorem for weighted sums of sequences of negatively dependent random variables. J Inequal Appl, Volume 2010, Article ID 383805, 8 p

  • Wu QY (2012) A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables. J Inequal Appl 2012:50

    Article  Google Scholar 

  • Yang WZ, Wang XJ, Wang XH, Hu SH (2012) The consistency for estimator of nonparametric regression model based on NOD errors. J Inequal Appl 2012:140

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors are most grateful to the Editor Norbert Henze and two anonymous reviewers for careful reading of the manuscript and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (11201001, 11171001, 11126176), the Natural Science Foundation of Anhui Province (1308085QA03, 1208085QA03, 1408085QA02), the Students Innovative Training Project of Anhui University (201410357118) and the Students Science Research Training Program of Anhui University (kyxl2013003).

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Authors and Affiliations

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

    Aiting Shen & Ying Zhang

  2. Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada

    Andrei Volodin

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  1. Aiting Shen
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  2. Ying Zhang
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  3. Andrei Volodin
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Correspondence to Aiting Shen.

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Shen, A., Zhang, Y. & Volodin, A. Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables. Metrika 78, 295–311 (2015). https://doi.org/10.1007/s00184-014-0503-y

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  • DOI: https://doi.org/10.1007/s00184-014-0503-y

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