Advertisement

Statistics and Computing

, Volume 17, Issue 2, pp 81–92 | Cite as

Robust mixture modeling using the skew t distribution

  • Tsung I. LinEmail author
  • Jack C. Lee
  • Wan J. Hsieh
Article

Abstract

A finite mixture model using the Student's t distribution has been recognized as a robust extension of normal mixtures. Recently, a mixture of skew normal distributions has been found to be effective in the treatment of heterogeneous data involving asymmetric behaviors across subclasses. In this article, we propose a robust mixture framework based on the skew t distribution to efficiently deal with heavy-tailedness, extra skewness and multimodality in a wide range of settings. Statistical mixture modeling based on normal, Student's t and skew normal distributions can be viewed as special cases of the skew t mixture model. We present analytically simple EM-type algorithms for iteratively computing maximum likelihood estimates. The proposed methodology is illustrated by analyzing a real data example.

Keywords

EM-type algorithms Maximum likelihood Outlying observations PX-EM algorithm Skew t mixtures Truncated normal 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Azzalini A. 1985. A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 12: 171–178.MathSciNetGoogle Scholar
  2. Azzalini A. 1986. Further results on a class of distributions which includes the normal ones. Statistica 46: 199–208.zbMATHMathSciNetGoogle Scholar
  3. Azzalini A. and Capitaino A. 2003. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. Journal of the Royal Statistical Society, Series B 65: 367–389.zbMATHCrossRefGoogle Scholar
  4. Basford K.E., Greenway D.R., McLachlan G.J., and Peel, D. 1997. Standard errors of fitted means under normal mixture. Computational Statistics 12: 1–17.zbMATHGoogle Scholar
  5. Dellaportas P. and Papageorgiou I. 2006. Multivariate mixtures of normals with unknown number of components. Statistics and Computing 16: 57–68.CrossRefMathSciNetGoogle Scholar
  6. Dempster A.P., Laird N.M., and Rubin D.B. 1977. Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, Series B 39: 1–38.zbMATHMathSciNetGoogle Scholar
  7. Flegal K.M., Carroll M.D., Ogden C.L., and Johnson C.L. 2002. Prevalence and trends in obesity among US adults, 1999–2000. Journal of the American Medical Association 288: 1723–1727.CrossRefGoogle Scholar
  8. Henze N. 1986. A probabilistic representation of the skew-normal distribution. Scandinavian Journal of Statistics 13: 271–275.MathSciNetGoogle Scholar
  9. Jones M.C. and Faddy M.J. 2003. A skew extension of the t-distribution, with applications. Journal of the Royal Statistical Society, Series B 65: 159–174.zbMATHCrossRefMathSciNetGoogle Scholar
  10. Lin T.I., Lee J.C., and Ni H.F. 2004. Bayesian analysis of mixture modelling using the multivariate t distribution. Statistics and Computing 14: 119–130.CrossRefMathSciNetGoogle Scholar
  11. Lin T.I., Lee J.C., and Yen S.Y. 2007. Finite mixture modelling using the skew normal distribution. Statistica Sinica (In press)Google Scholar
  12. Liu C.H. and Rubin D.B. 1994. The ECME algorithm: a simple extension of EM and ECM with faster monotone convergence. Biometrika 81: 633–648.zbMATHCrossRefMathSciNetGoogle Scholar
  13. Liu C.H., Rubin D.B., and Wu, Y. 1998. Parameter expansion to accelerate EM: the PX-EM algorithm. Biometrika 85: 755–770.zbMATHCrossRefMathSciNetGoogle Scholar
  14. McLachlan G.J. and Basford K.E. 1988. Mixture Models: Inference and Application to Clustering, Marcel Dekker, New York.Google Scholar
  15. McLachlan G.J. and Peel D. 2000. Finite Mixture Models, Wiely, New York.zbMATHCrossRefGoogle Scholar
  16. Meng X.L. and Rubin D.B. 1993. Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika 80:267–78.zbMATHCrossRefMathSciNetGoogle Scholar
  17. Peel D. and McLachlan G. J. 2000. Robust mixture modeling using the t distribution. Statistics and Computing 10: 339–348.CrossRefGoogle Scholar
  18. Richardson S. and Green P.J. 1997. On Bayesian analysis of mixtures with an unknown number of components (with discussion). Journal of the Royal Statistical Society, Series B 59: 731–792.zbMATHCrossRefMathSciNetGoogle Scholar
  19. Shoham S. 2002. Robust clustering by deterministic agglomeration EM of mixtures of multivariate t-distributions. Pattern Recognition 35: 1127–1142.zbMATHCrossRefGoogle Scholar
  20. Shoham S., Fellows M.R., and Normann R.A. 2003. Robust, automatic spike sorting using mixtures of multivariate t-distributions. Journal of Neuroscience Methods 127: 111–122.CrossRefGoogle Scholar
  21. Titterington D.M., Smith A.F.M., and Markov U.E. 1985. Statistical Analysis of Finite Mixture Distributions, Wiely, New York.zbMATHGoogle Scholar
  22. Wang H.X., Zhang Q.B., Luo B., and Wei S. 2004. Robust mixture modelling using multivariate t distribution with missing information. Pattern Recognition Letter 25: 701–710.CrossRefGoogle Scholar
  23. Zacks S. 1971. The Theory of Statistical Inference, New York, Wiley.Google Scholar
  24. Zhang Z., Chan K.L., Wu Y., and Cen C.B. 2004. Learning a multivariate Gaussian mixture model with the reversible Jump MCMC algorithm. Statistics and Computing 14: 343–355.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Applied-MathematicsNational Chung Hsing UniversityTaichungTaiwan
  2. 2.Graduate Institute of FinanceNational Chiao Tung UniversityHsinchuTaiwan
  3. 3.Institute of StatisticsNational Chiao Tung UniversityHsinchuTaiwan

Personalised recommendations