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Robust mixture modeling using multivariate skew t distributions

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Abstract

This paper presents a robust mixture modeling framework using the multivariate skew t distributions, an extension of the multivariate Student’s t family with additional shape parameters to regulate skewness. The proposed model results in a very complicated likelihood. Two variants of Monte Carlo EM algorithms are developed to carry out maximum likelihood estimation of mixture parameters. In addition, we offer a general information-based method for obtaining the asymptotic covariance matrix of maximum likelihood estimates. Some practical issues including the selection of starting values as well as the stopping criterion are also discussed. The proposed methodology is applied to a subset of the Australian Institute of Sport data for illustration.

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References

  • Arellano-Valle, R.B., Bolfarine, H., Lachos, V.H.: Bayesian inference for skew-normal linear mixed models. J. Appl. Stat. 34, 663–682 (2007)

    Article  MathSciNet  Google Scholar 

  • Azzalini, A.: The skew-normal distribution and related multivariate families (with discussion). Scand. J. Statist. 32, 159–200 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Azzalini, A., Capitaino, A.: Statistical applications of the multivariate skew-normal distribution. J. R. Stat. Soc. Ser. B 61, 579–602 (1999)

    Article  MATH  Google Scholar 

  • Azzalini, A., Capitaino, A.: Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. J. R. Stat. Soc. Ser. B 65, 367–389 (2003)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Basford, K.E., Greenway, D.R., McLachlan, G.J., Peel, D.: Standard errors of fitted means under normal mixture. Comput. Stat. 12, 1–17 (1997)

    MATH  Google Scholar 

  • Booth, G.J., Hobert, P.J.: Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. J. R. Stat. Soc. Ser. B 61, 265–285 (1999)

    Article  MATH  Google Scholar 

  • Cook, R.D., Weisberg, S.: An Introduction to Regression Graphics. Wiley, New York (1994)

    Book  MATH  Google Scholar 

  • Dellaportas, P., Papageorgiou, I.: Multivariate mixtures of normals with unknown number of components. Stat. Comput. 16, 57–68 (2006)

    Article  MathSciNet  Google Scholar 

  • Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. R. Stat. Soc. Ser. B 39, 1–38 (1977)

    MATH  MathSciNet  Google Scholar 

  • Diebolt, J., Robert, C.P.: Estimation of finite mixture distributions through Bayesian sampling. J. R. Stat. Soc. Ser. B 56, 363–375 (1994)

    MATH  MathSciNet  Google Scholar 

  • Escobar, M.D., West, M.: Bayesian density estimation and inference using mixtures. J. Am. Stat. Assoc. 90, 577–588 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Fraley, C., Raftery, A.E.: How many clusters? Which clustering method? Answers via model-based cluster analysis. Comput. J. 41, 578–588 (1998)

    MATH  Google Scholar 

  • Fraley, C., Raftery, A.E.: Model-based clustering, discriminant analysis, and density estimation. J. Am. Stat. Assoc. 97, 611–612 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Frühwirth-Schnatter, S.: Finite Mixture and Markov Switching Models. Springer, New York (2006)

    MATH  Google Scholar 

  • Keribin, C.: Consistent estimation of the order of mixture models. Sankhyā Ser. 62, 49–66 (2000)

    MATH  MathSciNet  Google Scholar 

  • Lin, T.I.: Maximum likelihood estimation for multivariate skew normal mixture models. J. Multivar. Anal. 100, 257–265 (2009)

    Article  MATH  Google Scholar 

  • Lin, T.I., Lee, J.C., Hsieh, W.J.: Robust mixture modeling using the skew t distribution. Stat. Comput. 17, 81–92 (2007a)

    Article  MathSciNet  Google Scholar 

  • Lin, T.I., Lee, J.C., Yen, S.Y.: Finite mixture modelling using the skew normal distribution. Stat. Sin. 17, 909–927 (2007b)

    MATH  MathSciNet  Google Scholar 

  • Lindsay, B.: Mixture Models: Theory, Geometry and Applications. Institute of Mathematical Statistics, Hayward (1995)

    MATH  Google Scholar 

  • Liu, C.H., Rubin, D.B.: The ECME algorithm: a simple extension of EM and ECM with faster monotone convergence. Biometrika 81, 633–648 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  • Lo, K., Brinkman, R.R., Gottardo, R.: Automated gating of flow cytometry data via robust model-based clustering. Cytometry Part A 73, 321–332 (2008)

    Article  Google Scholar 

  • Louis, T.A.: Finding the observed information when using the EM algorithm. J. R. Stat. Soc. Ser. B 44, 226–232 (1982)

    MATH  MathSciNet  Google Scholar 

  • McCulloch, C.E.: Maximum likelihood variance components estimation for binary data. J. Am. Stat. Assoc. 89, 330–335 (1994)

    Article  MATH  Google Scholar 

  • McLachlan, G.J., Basford, K.E.: Mixture Models: Inference and Application to Clustering. Dekker, New York (1988)

    Google Scholar 

  • McLachlan, G.J., Krishnan, T.: The EM Algorithm and Extensions, 2nd edn. Wiley, New York (2008)

    MATH  Google Scholar 

  • McLachlan, G.J., Peel, D.: Finite Mixture Models. Wiley, New York (2000)

    Book  MATH  Google Scholar 

  • McNicholas, P.D., Murphy, T.B.: Parsimonious Gaussian mixture models. Stat. Comput. 18, 285–296 (2008)

    Article  MathSciNet  Google Scholar 

  • Meilijson, I.: A fast improvement to the EM algorithm to its own terms. J. R. Stat. Soc. Ser. B 51, 127–138 (1989)

    MATH  MathSciNet  Google Scholar 

  • Meng, X.L., Rubin, D.B.: Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika 80, 267–278 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  • Nadarajah, S., Kotz, S.: Programs in R for computing truncated t distributions. Qual. Reliab. Eng. Int. 23, 273–278 (2007)

    Article  Google Scholar 

  • Peel, D., McLachlan, G.J.: Robust Mixture modeling using the t distribution. Stat. Comput. 10, 339–348 (2000)

    Article  Google Scholar 

  • Pyne, S., Hu, X., Wang, K., Rossin, E., Lin, T.I., Maier, L., Baecher-Allan, C., McLachlan, G.J., Tamayo, P., Hafler, D.A., De Jager, P.L., Mesirov, J.P.: Automated high-dimensional flow cytometric data analysis. Proc. Natl. Acad. Sci. USA (2009). doi:10.1073/pnas.0903028106

    Google Scholar 

  • Redner, R.A., Walker, H.F.: Mixture densities, maximum likelihood and the EM algorithm. SIAM Rev. 26, 195–239 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  • R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2008)

    Google Scholar 

  • Richardson, S., Green, P.J.: On Bayesian analysis of mixtures with an unknown number of components (with discussion). J. R. Stat. Soc. Ser. B 59, 731–792 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • Sahu, S.K., Dey, D.K., Branco, M.D.: A new class of multivariate skew distributions with application to Bayesian regression models. Can. J. Stat. 31, 129–150 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Titterington, D.M., Smith, A.F.M., Markov, U.E.: Statistical Analysis of Finite Mixture Distributions. Wiley, New York (1985)

    MATH  Google Scholar 

  • Wei, G.C.G., Tanner, M.A.: A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithms. J. Am. Stat. Assoc. 85, 699–704 (1990)

    Article  Google Scholar 

  • Zhang, Z., Chan, K.L., Wu, Y., Cen, C.B.: Learning a multivariate Gaussian mixture model with the reversible Jump MCMC algorithm. Stat. Comput. 14, 343–355 (2004)

    Article  MathSciNet  Google Scholar 

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  1. Department of Applied Mathematics and Institute of Statistics, National Chung Hsing University, Taichung, 402, Taiwan

    Tsung-I Lin

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Lin, TI. Robust mixture modeling using multivariate skew t distributions. Stat Comput 20, 343–356 (2010). https://doi.org/10.1007/s11222-009-9128-9

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