Skip to main content
SpringerLink
Log in

Inferences in linear mixed models with skew-normal random effects

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

For the linear mixed model with skew-normal random effects, this paper gives the density function, moment generating function and independence conditions. The noncentral skew chi-square distribution is defined and its density function is shown. The necessary and sufficient conditions under which a quadratic form is distributed as noncentral skew chi-square distribution are obtained. Also, a version of Cochran’s theorem is given, which modifies the result of Wang et al. (2009) and is used to set up exact tests for fixed effects and variance components of the proposed model. For illustration, our main results are applied to a real data problem.

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

Similar content being viewed by others

References

  1. Afriat, S. N.: Orthogonal and oblique projectors and the characteristics of pairs of vector spaces. Proc. Camb. Philos. Soc., 53, 800–816 (1957)

    Article  MathSciNet  Google Scholar 

  2. Arellano-Valle, R. B., Bolfarine, H., Lachos, V. H.: Skew-normal linear mixed models. J. Data Science, 3, 415–438 (2005)

    Google Scholar 

  3. Azzalini, A.: The skew-normal distribution and related multivariate families. Scand. J. Statist., 32, 159–188 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Azzalini, A., Capitanio, A.: Statistical applications of the multivariate skew normal distributions. J. Roy. Statist. Soc. Ser. B, 61, 579–602 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Azzalini, A., Dalla Valle, A.: The multivariate skew-normal distribution. Biometrika, 83, 715–726 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  6. Baksalary, J. K.: Algebraic Characterizations and Statistical Implications of the Commutativity of Orthogonal Projectors. In: Proceedings of Second International Tampere Conference in Statistics (Edited by S. Puntanen, T. Pukkila), 1987, 113–142

    Google Scholar 

  7. Baltagi, B. H.: Econometric Analysis of Panel Data, Wiley, New York, 1995

    MATH  Google Scholar 

  8. Diggle, P. J., Heagerty, P., Liang, K. E., et al.: Analysis of Longitudinal Data, Oxford Science, New York, 2002

    Google Scholar 

  9. Ghidey, W., Lesaffre, E., Eilers, P.: Smooth random effects distribution in a linear mixed model. Biometrics, 60, 945–953 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lachos, V. H., Bolfarine, H., Arellano-Valle, R. B., et al.: Likelihood-based inference for multivariate skew-normal regression models. Comm. Statist. Theory Methods, 36, 1769–1786 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lachos, V. H., Dey, D. K., Cancho, V. G.: Robust linear mixed models with skew-normal independent distributions from a Bayesian perspective. J. Statist. Plann. Inference, 139, 4098–4110 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lachos, V. H., Ghosh, P., Arellano-Valle, R. B.: Likelihood based inference for skew-normal independent linear mixed models. Statist. Sinica, 20, 303–322 (2010)

    MATH  MathSciNet  Google Scholar 

  13. Laird, N. M., Ware, J. H.: Random effects models for longitudinal data. Biometrics, 38, 963–974 (1982)

    Article  MATH  Google Scholar 

  14. Lin, T. I., Lee, J. C.: Estimation and prediction in linear mixed models with skew-normal random effects for longitudinal data. Statist. Med., 27, 1490–1507 (2008)

    Article  MathSciNet  Google Scholar 

  15. Ma, Y., Genton, M. G., Davidian, M.: Linear Mixed Models with Flexible Generalized Skew-Elliptical Random Effects. In: Skew-Elliptical Distributions and their Applications: A Journey Beyond Normality, Edited by M. G. Genton, Chapman & Hall/CRC, Boca Raton, FL, 2004, 339–358

    Google Scholar 

  16. Mangder, L. S., Zeger, S. L.: A smooth nonparametric estimate of a mixing distribution using mixtures of Gaussians. J. Amer. Statist. Assoc., 91, 1141–1151 (1996)

    Article  MathSciNet  Google Scholar 

  17. Muirhead, R. J.: Aspects of Multivariate Statistical Theory, Wiley, New York, 1982

    Book  MATH  Google Scholar 

  18. Pinheiro, J. C., Liu, C. H., Wu, Y. N.: Efficient algorithms for robust estimation in linear mixed-effects models using a multivariate t-distribution. J. Comput. Graph. Statist., 10, 249–276 (2001)

    Article  MathSciNet  Google Scholar 

  19. Tao, H., Palta, M., Yandell, B. S., et al.: An estimation method for the semiparametric mixed effects model. Biometrics, 55, 102–110 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  20. Tjur, T.: Analysis of variance models in orthogonal designs. Int. Stat. Rev., 52, 33–81 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  21. Verbeke, G., Lesaffre, E.: The effect of misspecifying the random effects distribution in linear mixed models for longitudinal data. Comput. Statist. Data Anal., 23, 541–556 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  22. Wang, T., Li, B., Gupta, A. K.: Distribution of quadratic forms under skew normal settings. J. Multivariate Anal., 100, 533–545 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Wu, L.: Mixed Effects Models for Complex Data, Chapman & Hall/CRC, Boca Raton, 2010

    MATH  Google Scholar 

  24. Wu, M. X.: Introduction for Linear Mixed Effects Models, Science Press, Beijing, 2013

    Google Scholar 

  25. Ye, R. D., Wang, T., Gupta, A. K.: Distribution of matrix quadratic forms under skew-normal settings. J. Multivariate Anal., 131, 229–239 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  26. Zhang, D., Davidian, M.: Linear mixed models with flexible distributions of random effects for longitudinal data. Biometrics, 57, 795–802 (2001)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Economics, Hangzhou Dianzi University, Zhejiang, 310018, P. R. China

    Ren Dao Ye

  2. Innovation Experimental College, Northwest A&F University, Shannxi, 712100, P. R. China

    Tong Hui Wang

  3. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, 88003, USA

    Tong Hui Wang

Authors
  1. Ren Dao Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tong Hui Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tong Hui Wang.

Additional information

This work is done during the visit of the first author to New Mexico State University. This work is supported by National Natural Science Foundation of China (Grant No. 11401148), Ministry of Education of China, Humanities and Social Science Projects (Grant Nos. 14YJC910005, 10YJC790184), Zhejiang Provincial Natural Science Foundation of China (Grant No. LY14A010030), Zhejiang Provincial Philosophy and Social Science Planning Project of China (Grant No. 13NDJC089YB), and Houji Scholar Fund of Northwest A and F University, China

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ye, R.D., Wang, T.H. Inferences in linear mixed models with skew-normal random effects. Acta. Math. Sin.-English Ser. 31, 576–594 (2015). https://doi.org/10.1007/s10114-015-3326-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-015-3326-5

Keywords

MR(2010) Subject Classification

Search

Navigation