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On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models

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Abstract

In this paper, some probability inequalities and moment inequalities for widely orthant-dependent (WOD, in short) random variables are presented, especially the Marcinkiewicz–Zygmund type inequality and Rosenthal type inequality. By using these inequalities, we further study the complete convergence for weighted sums of arrays of row-wise WOD random variables and give some special cases, which extend some corresponding ones for dependent sequences. As applications, we present some sufficient conditions to prove the complete consistency for the estimator of nonparametric regression model based on WOD errors by using the complete convergence that we established. At last, the choice of the fixed design points and the weight functions for the nearest neighbor estimates is proposed. Our results generalize some known results for independent random variables and some dependent random variables.

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Acknowledgments

The authors are most grateful to the Editor-in-Chief Alfonso Gordaliza, Associated Editor, and two anonymous referees 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 Science, Anhui University, Hefei, 230601, People’s Republic of China

    Xuejun Wang, Chen Xu & Shuhe Hu

  2. Department of Mathematics, Tsing Hua University, Hsinchu, Taiwan, 30043, Taiwan, ROC

    Tien-Chung Hu

  3. Department of Mathematics and Statistics, University of Regina, Regina, SK, S4S 0A2, Canada

    Andrei Volodin

Authors
  1. Xuejun Wang
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  2. Chen Xu
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  3. Tien-Chung Hu
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  4. Andrei Volodin
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  5. Shuhe Hu
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Corresponding author

Correspondence to Xuejun Wang.

Additional information

This work was supported by the National Natural Science Foundation of China (11201001, 11171001, 11126176), the Natural Science Foundation of Anhui Province (1208085QA03, 1308085QA03) and Doctoral Scientific Research Foundation of Anhui University.

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Wang, X., Xu, C., Hu, TC. et al. On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. TEST 23, 607–629 (2014). https://doi.org/10.1007/s11749-014-0365-7

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  • DOI: https://doi.org/10.1007/s11749-014-0365-7

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