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Complete convergence for weighted sums of NSD random variables and its application in the EV regression model

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Abstract

In this paper, some basic properties for negatively superadditive-dependent (NSD, in short) random variables are presented, such as the Rosenthal-type inequality and the Kolmogorov-type exponential inequality. Using these properties, we further study the complete convergence for weighted sums of NSD random variables, which generalizes and improves some corresponding ones for independent random variables and negatively associated random variables. Some sufficient conditions to prove the complete convergence for weighted sums of NSD random variables are provided. As an application, the complete consistency of LS estimators in the EV regression model with NSD errors is investigated under mild conditions, which generalizes and improves the corresponding one for negatively associated random variables.

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References

  • Baum LE, Katz M (1965) Convergence rates in the law of large numbers. T Am Math Soc 120:108–123

    Article  MATH  MathSciNet  Google Scholar 

  • Carroll RJ, Ruppert D, Stefanski LA (1995) Measurement error in nonlinear models. Chapman and Hall, New York

    Book  MATH  Google Scholar 

  • 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 

  • Cui HJ (1997) Asymptotic normality of \(M\)-estimates in the EV model. J Syst Sci Complex 10:225–236

    MATH  Google Scholar 

  • Deaton A (1985) Panel data from a time series of cross-sections. J Econ 30:109–126

    Article  MATH  Google Scholar 

  • Eghbal N, Amini M, Bozorgnia A (2010) Some maximal inequalities for quadratic forms of negative superadditive dependence 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 

  • Erdös P (1949) On a theorem of Hsu and Robbins. Ann Stat 20:286–291

    Article  MATH  Google Scholar 

  • Fan GL, Liang HY, Wang JF, Xu HX (2010) Asymptotic properties for LS estimators in EV regression model with dependent errors. AStA Adv Stat Anal 94:89–103

    Article  MathSciNet  Google Scholar 

  • Fazekas I, Kukush AG (1997) Asymptotic properties of an estimator in nonlinear functional errors-in-variables models with dependent error terms. Comput Math Appl 34:23–39

    Article  MATH  MathSciNet  Google Scholar 

  • Fuller WA (1987) Measurement error models. Wiley, New York

    Book  MATH  Google Scholar 

  • Fusek I, Fusková L (1989) A combined estimator in the simple errors-in-variables model. Stat Pap 30:1–15

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Hu TZ, Xie CD, Ruan LY (2005) Dependence structures of multivariate Bernoulli random vectors. J Multivar Anal 94:172–195

    Article  MATH  MathSciNet  Google Scholar 

  • Hslao C, Wang L, Wang Q (1997) Estimation of nonlinear errors-in-variables models: an approximate solution. Stat Pap 38:1–25

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  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 

  • Li DL, Rao MB, Jiang TF, Wang XC (1995) Complete convergence and almost sure convergence of weighted sums of random variables. J Theor Probab 8:49–76

    Article  MATH  MathSciNet  Google Scholar 

  • Liu JX, Chen XR (2005) Consistency of LS estimator in simple linear EV regression models. Acta Math Sci 25:50–58

    MATH  Google Scholar 

  • Miao Y, Yang GY, Shen LM (2007) The central limit theorem for LS estimator in simple linear EV regression models. Commun Stat Theory Methods 36:2263–2272

    Article  MATH  MathSciNet  Google Scholar 

  • Miao Y, Wang K, Zhao FF (2011) Some limit behaviors for the LS estimator in simple linear EV regression models. Stat Probab Lett 81:92–102

    Article  MATH  MathSciNet  Google Scholar 

  • Miao Y, Zhao FF, Wang K, Chen YP (2013) Asymptotic normality and strong consistency of LS estimators in the EV regression model with NA errors. Stat Pap 54:193–206

    Article  MATH  MathSciNet  Google Scholar 

  • Mittag HJ (1989) Estimating parameters in a simple errors-in-variables model: a new approach based on finite sample distribution theory. Stat Pap 30:133–140

    Article  Google Scholar 

  • Shao QM (2000) A comparison theorem on moment inequalities between negatively associated and independent random variables. J Theor Probab 13:343–355

    Article  MATH  Google Scholar 

  • Shen AT (2013) On the strong convergence rate for weighted sums of arrays of rowwise negatively orthant dependent random variables. RACSAM 107:257–271

    Article  MATH  Google Scholar 

  • Shen AT, Wu RC (2013) Strong and weak convergence for asymptotically almost negatively associated random variables. Discret Dyn Nat Soc 2013, Article ID 235012. doi:10.1155/2013/235012

  • Shen AT, Wu RC, Wang XH, Shen Y (2013a) Complete convergence for weighted sums of arrays of rowwise \(\tilde{\rho }\)-mixing random variables. J Inequal Appl 2013, Article ID 356. doi:10.1186/1029-242X-2013-356

  • Shen AT, Wu RC, Chen Y, Zhou Y (2013b) Complete convergence of the maximum partial sums for arrays of rowwise of AANA random variables. Discret Dyn Nat Soc 2013, Article ID 741901. doi:10.1155/2013/741901

  • Spitzer FL (1956) A combinatorial lemma and its application to probability theory. T Am Math Soc 82:323–339

    Article  MATH  MathSciNet  Google Scholar 

  • Sung SH (2010) Complete convergence for weighted sums of \(\rho ^*\)-mixing random variables. Discret Dyn Nat Soc 2010, Article ID 630608. doi:10.1155/2010/630608

  • Sung SH (2012) A note on the complete convergence for weighted sums of negatively dependent random variables. J Inequal Appl 2012, Article ID 158. doi:10.1186/1029-242X-2012-158

  • Thrum R (1987) A remark on almost sure convergence of weighted sums. Probab Theory Rel 75:425–430

    Article  MATH  MathSciNet  Google Scholar 

  • Wang XJ, Li XQ, Yang WZ, Hu SH (2012a) On complete convergence for arrays of rowwise weakly dependent random variables. Appl Math Lett 25:1916–1920

    Article  MATH  MathSciNet  Google Scholar 

  • Wang XJ, Hu SH, Yang WZ (2012b) Complete convergence for arrays of rowwise negatively orthant dependent random variables. RACSAM 106:235–245

    Article  MATH  MathSciNet  Google Scholar 

  • Wang XJ, Deng X, Zheng LL, Hu SH (2014a) Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications. Stat J Theor Appl Stat 48:834–850

    MATH  MathSciNet  Google Scholar 

  • Wang XJ, Xu C, Hu TC, Volodin A, Hu SH (2014b) On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. TEST 23:607–629

    Article  MATH  MathSciNet  Google Scholar 

  • Wu QY (2006) Probability limit throey for mixing sequences. Science Press of China, Beijing

    Google Scholar 

  • Wu QY (2010) Complete convergence for negatively dependent sequences of random variables. J Inequal Appl 2010, Article ID 507293. doi:10.1155/2010/507293

  • Wu QY (2012a) A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables. J Inequal Appl 2012, Article ID 50. doi:10.1186/1029-242X-2012-50

  • Wu QY (2012b) Sufficient and necessary conditions of complete convergence for weighted sums of PNQD random variables. J Appl Math 2012, Article ID 104390. doi:10.1155/2012/104390

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Acknowledgments

The authors are most grateful to the Editor-in-Chief Ana F. Militino, Associate Editor and three anonymous reviewers for careful reading of 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

    Xuejun Wang, Aiting Shen, Zhiyong Chen & Shuhe Hu

Authors
  1. Xuejun Wang
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  2. Aiting Shen
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  3. Zhiyong Chen
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  4. Shuhe Hu
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Correspondence to Aiting Shen.

Additional information

This work was supported by the National Natural Science Foundation of China (11201001, 11171001, 11126176), the Natural Science Foundation of Anhui Province (1308085QA03, 1408085QA02), the Research Teaching Model Curriculum of Anhui University (xjyjkc1407) and the Students Innovative Training Project of Anhui University (201410357117, 201410357249).

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Wang, X., Shen, A., Chen, Z. et al. Complete convergence for weighted sums of NSD random variables and its application in the EV regression model. TEST 24, 166–184 (2015). https://doi.org/10.1007/s11749-014-0402-6

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  • DOI: https://doi.org/10.1007/s11749-014-0402-6

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