Skip to main content
SpringerLink
Log in

Complete moment convergence for negatively orthant dependent random variables and its applications in statistical models

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

In this paper, a general result on complete moment convergence for arrays of rowwise negatively orthant dependent random variables is obtained. In addition, we present some sufficient conditions to prove the complete moment and complete convergences for the variables. As applications, the complete consistency for the estimators of nonparametric and semiparametric regression models based on negatively orthant dependent errors is established by using the complete convergence that we established. A simulation to study the numerical performance of the consistency for the nearest neighbor weight function estimator in semiparametric regression model is given. Our results generalize and improve some corresponding ones for independent random variables and negatively associated random variables.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Amini M, Azarnoosh HA, Bozorgnia A (2004) The strong law of large numbers for negatively dependent generalized Gaussian random variables. Stoch Anal Appl 22:893–901

    Article  MathSciNet  Google Scholar 

  • Asadian N, Fakoor V, Bozorgnia A (2006) Rosenthal’s type inequalities for negatively orthant dependent random variables. J Iranian Stat Soc 5(1–2):66–75

    MATH  Google Scholar 

  • Bai ZD, Su C (1985) The complete convergence for partial sums of i.i.d. random variables. Sci China Ser A 28(12):1261–1277

    MathSciNet  MATH  Google Scholar 

  • Bozorgnia A, Patterson RF, Taylor RL (1996) Limit theorems for dependent random variables. In: Proceedings of the First World Congress of Nonlinear Analysts. Gruyter Publishers, Berlin, pp. 1639–1650

  • Chow YS (1988) On the rate of moment complete convergence of sample sums and extremes. Bull Inst Math Acad Sin 16(3):177–201

    MATH  Google Scholar 

  • Erdös P (1949) On a theorem of Hsu and Robbins. Ann Math Stat 20(2):286–291

    Article  MathSciNet  Google Scholar 

  • Fan Y (1990) Consistent nonparametric multiple regression for dependent heterogeneous processes. J Multivar Anal 33(1):72–88

    Article  Google Scholar 

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

  • Georgiev AA (1988) Consistent nonparametric multiple regression: the fixed design case. J Multivar Anal 25(1):100–110

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Hsu PL, Robbins H (1947) Complete convergence and the law of large numbers. Proc Natl Acad Sci USA 33(2):25–31

    Article  MathSciNet  Google Scholar 

  • Hu SH (1999) Estimator for a semiparametric regression model. Acta Math Sci Ser A 19:541–549

    Article  Google Scholar 

  • Hu SH (2006) Fixed-design semiparametric regression for linear time series. Acta Math Sci Ser B 26(1):74–82

    Article  MathSciNet  Google Scholar 

  • Hu SH, Zhu CH, Chen YB, Wang LC (2002) Fixed-design regression for linear time series. Acta Math Sci Ser B 22(1):9–18

    Article  MathSciNet  Google Scholar 

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

    Article  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(1):49–76

    Article  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  MathSciNet  Google Scholar 

  • Müller HG (1987) Weak and universal consistency of moving weighted averages. Periodica Math Hung 18(3):241–250

    Article  MathSciNet  Google Scholar 

  • Pan GM, Hu SH, Fang LB, Cheng ZD (2003) Mean consistency for a semiparametric regression model. Acta Math Sci Ser A 23(5):598–606

    MathSciNet  MATH  Google Scholar 

  • Qiu DH, Chang KC, Antoninic RG, Volodin A (2011) On the strong rates of convergence for arrays of rowwise negatively dependent random variables. Stoch Anal Appl 29:375–385

    Article  MathSciNet  Google Scholar 

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

    Article  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  MathSciNet  Google Scholar 

  • Shen AT (2011) Some strong limit theorems for arrays of rowwise negatively orthant-dependent random variables. J Inequal Appl 2011(93):10

    MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Shen AT (2013b) Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models. Abstr Appl Anal 2013(862602):9

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

    Article  MathSciNet  Google Scholar 

  • Sung SH (2011) On the exponential inequalities for negatively dependent random variables. J Math Anal Appl 381:538–545

    Article  MathSciNet  Google Scholar 

  • Sung SH (2012) Complete convergence for weighted sums of negatively dependent random variables. Stat Pap 53:73–82

    Article  MathSciNet  Google Scholar 

  • Taylor RL, Patterson RF, Bozorgnia A (2002) A strong law of large numbers for arrays of rowwise negatively dependent random variables. Stoch Anal Appl 20:643–656

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Wang DC, Zhao W (2006) Moment complete convergence for sums of a sequence of NA random variables. Appl Math 21(4):445–450

    MathSciNet  MATH  Google Scholar 

  • Wang XJ, Hu SH, Yang WZ, Ling NX (2010) Exponential inequalities and inverse moment for NOD sequence. Stat Probab Lett 80(5–6):452–461

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Wang XJ, Deng X, Zheng LL, Hu SH (2014) Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications. Statistics 48(4):834–850

    Article  MathSciNet  Google Scholar 

  • Wang XJ, Chen ZY, Xiao R, Xie XJ (2017) Complete moment convergence for weighted sums of negatively orthant dependent random variables. Filomat 31(5):1195–1206

    Article  MathSciNet  Google Scholar 

  • Wu QY (2006) Probability Limiting Theory for Mixing Sequences. Science Press of China, Beijing

    Google Scholar 

  • Wu QY (2011) Complete convergence for weighted sums of sequences of negatively dependent random variables. J Probab Stat 2011(202015):16

    MathSciNet  MATH  Google Scholar 

  • Wu QY, Jiang YY (2011) The strong consistency of \(M\) estimator in a linear model for negatively dependent random samples. Commun Stat Theory Methods 40:467–491

    Article  MathSciNet  Google Scholar 

  • Wu YF (2014) On complete moment convergence for arrays of rowwise negatively associated random variables. RACSAM 108(2):669–681

    Article  MathSciNet  Google Scholar 

  • Yan ZZ, Wu WZ, Zie ZK (2001) Near neighbor estimate in Semiparametric regression model: the martingale difference error sequence case. J Appl Probab Stat 17:44–50

    MathSciNet  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):13

    MathSciNet  MATH  Google Scholar 

  • Zarei H, Jabbari H (2011) Complete convergence of weighted sums under negative dependence. Stat Pap 52:413–418

    Article  MathSciNet  Google Scholar 

  • Zhou XC, Lin JG (2013) Asymptotic properties of wavelet estimators in semiparametric regression models under dependent errors. J Multivar Anal 122:251–270

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

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

    Xuejun Wang, Yi Wu & Shuhe Hu

  2. School of Mathematics, Hefei University of Technology, Hefei, 230601, People’s Republic of China

    Nengxiang Ling

Authors
  1. Xuejun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shuhe Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nengxiang Ling
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuejun Wang.

Additional information

Supported by the National Natural Science Foundation of China (11671012, 11501004, 11501005), the National Social Science Foundation of China (14ATJ005), the Natural Science Foundation of Anhui Province (1508085J06) and the Key Projects for Academic Talent of Anhui Province (gxbjZD2016005).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, X., Wu, Y., Hu, S. et al. Complete moment convergence for negatively orthant dependent random variables and its applications in statistical models. Stat Papers 61, 1147–1180 (2020). https://doi.org/10.1007/s00362-018-0983-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-018-0983-3

Keywords

Mathematics Subject Classification

Search

Navigation