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Asymptotic Properties of Maximum Quasi-Likelihood Estimators in Generalized Linear Models with Diverging Number of Covariates

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Abstract

In this paper, for the generalized linear models (GLMs) with diverging number of covariates, the asymptotic properties of maximum quasi-likelihood estimators (MQLEs) under some regular conditions are developed. The existence, weak convergence and the rate of convergence and asymptotic normality of linear combination of MQLEs and asymptotic distribution of single linear hypothesis test statistics are presented. The results are illustrated by Monte-Carlo simulations.

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References

  1. Nelder J A and Wedderburm RWM, Generalized models, Journal of the Royal Statistical Society: Series B, 1972, 135: 370–384.

    Article  Google Scholar 

  2. Wedderburm R W M, Quasi-likelihood functions, generalized linear models, and the Gauss-Newton, Biometrika, 1974, 61: 439–447.

    Google Scholar 

  3. Liang K Y and Zeger S L, Longitudinal data analysis using generalized linear models, Biometrika, 1986, 73(1): 13–22.

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen K, Hu I, and Ying Z, Strong consistency of maximum quasi-likelihood estimators in generalized linear models with fixed and adaptive design, The Annals of Statistics, 1999, 27: 1155–1163.

    Article  MathSciNet  MATH  Google Scholar 

  5. Yue L and Chen X R, Rate of strong consistency of quasi-maximum likelihood estimate in generalized linear models, Science in China Series A Mathematics, 2004, 47(6): 882–803.

    Article  MathSciNet  MATH  Google Scholar 

  6. Yin C M and Zhao L C, Asymptotic normality and strong consistency of maximum quasilikelihhod in generalized linear models, Science in China Series A Mathematics, 2006, 49: 145–157.

    Article  MathSciNet  MATH  Google Scholar 

  7. Liao Y, Zhang S, and Xue H, Weak consistency of quasi-likelihood estimates in multivariate generalized linear models, Chinese Journal of Applied Probability and Statistics, 2006, 22(3): 288–294.

    MathSciNet  MATH  Google Scholar 

  8. Gao Q B, Wu Y H, Zhu C H, et al., Asymptotic normality of maximum quasi-likelihood estimators in generalized linear models with fixed design, Journal of Systems Science and Comlexity, 2008, 21(3): 463–473.

    Article  MathSciNet  MATH  Google Scholar 

  9. Gao Q B, Lin J G, Wu Y H, et al., Asymptotic normality of maximum quasi-likelihood estimators in generalized linear models with adaptive design, Statistics, 2012, 46(6): 833–846.

    Article  MathSciNet  MATH  Google Scholar 

  10. Xie M and Yang Y, Asymptotics for generalized estimating equations with large cluster sizes, The Annals of Statistics, 2003, 31: 310–347.

    Article  MathSciNet  MATH  Google Scholar 

  11. Huber P J, Robust regression: Asymptotics, conjectures and Monte Carlo, The Annals of Statistics, 1973, 1: 799–821.

    MATH  Google Scholar 

  12. Yohai V J and Maronna R A, Asymptotic behavior of M-estimators for the linear model, The Annals of Statistics, 1979, 7: 258–268.

    Article  MathSciNet  MATH  Google Scholar 

  13. Portnoy S, Asymptotic behavior of M-estimators of p regression parameters when p2/n is large I Consistency, The Annals of Statistics, 1984, 12: 1298–1309.

    Article  MathSciNet  MATH  Google Scholar 

  14. Portnoy S, Asymptotic behavior of M-estimators of p regression parameters when p2/n is large II Normal approximation, The Annals of Statistics, 1985, 13: 1403–1417.

    Article  MathSciNet  MATH  Google Scholar 

  15. Mammen E, Asymptotics with increasing dimension for robust regression with applications to the bootstrap, The Annals of Statistics, 1989, 17: 382–400.

    Article  MathSciNet  MATH  Google Scholar 

  16. Welsh A H, On M-processes and M-estimation, The Annals of Statistics, 1989, 17: 337–361.

    Article  MathSciNet  MATH  Google Scholar 

  17. Bai Z and Wu Y, Limiting behavior of M-estimators of regression coefficients in high dimensional linear models I. Scale-dependent case, Journal of Multivariate Analysis, 1994, 51: 211–239.

    Article  MathSciNet  MATH  Google Scholar 

  18. He X M and Shao Q M, On parameters of increasing dimensions, Journal of Multivariate Analysis, 2000, 73: 120–135.

    Article  MathSciNet  MATH  Google Scholar 

  19. Li G, Peng H, and Zhu L, Nonconcave penalized M-estimation with a diverging number of parameters, Statistica Sinica, 2011, 21: 391–419.

    MathSciNet  MATH  Google Scholar 

  20. Wang L, GEE analysis of clustered binary data with diverging number of covariates, The Annals of Statistics, 2011, 39: 389–417.

    Article  MathSciNet  MATH  Google Scholar 

  21. Ortega J and Rheinboldt W, Iterative Solution of Nonlinear Equation in Several Variables, Academic Press, New York, 1970.

    MATH  Google Scholar 

  22. Heuser H, Lehrbuch der Analysis, Teil 2, Stuttgart: Teubner, 1991.

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics Science, Nanjing Normal University, Nanjing, 210023, China

    Qibing Gao, Xiuli Du, Xiuqing Zhou & Fengchang Xie

Authors
  1. Qibing Gao
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  2. Xiuli Du
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  3. Xiuqing Zhou
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  4. Fengchang Xie
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Corresponding author

Correspondence to Qibing Gao.

Additional information

This paper was supported by Major Programm of Natural Science Foundation of China under Grant No. 71690242, the Natural Science Foundation of China under Grant No. 11471252, and the National Social Science Fund of China under Grant No. 18BTJ040.

This paper was recommended for publication by Editor DONG Yuexiao.

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Gao, Q., Du, X., Zhou, X. et al. Asymptotic Properties of Maximum Quasi-Likelihood Estimators in Generalized Linear Models with Diverging Number of Covariates. J Syst Sci Complex 31, 1362–1376 (2018). https://doi.org/10.1007/s11424-018-7017-z

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  • DOI: https://doi.org/10.1007/s11424-018-7017-z

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