ECM Algorithm for Fitting of Mixtures of Multivariate Skew t-Distribution

  • Kim, Seung-Gu (Department of Data and Information, Sangji University)
  • Received : 2012.06.10
  • Accepted : 2012.07.19
  • Published : 2012.09.30


Cabral et al. (2012) defined a mixture model of multivariate skew t-distributions(STMM), and proposed the use of an ECME algorithm (a variation of a standard EM algorithm) to fit the model. Their estimation by the ECME algorithm is closely related to the estimation of the degree of freedoms in the STMM. With the ECME, their purpose is to escape from the calculation of a conditional expectation that is not provided by a closed form; however, their estimates are quite unstable during the procedure of the ECME algorithm. In this paper, we provide a conditional expectation as a closed form so that it can be easily calculated; in addition, we propose to use the ECM algorithm in order to stably fit the STMM.


Supported by : Sangji University


  1. Azzalini, A. (1985). A class of distribution which includes the normal ones, Scandinavian Journal of Statistics, 33, 561-574.
  2. Azzalini, A. and Dalla-Valle, A. (1996). The multivariate skew normal distribution, Biometrika, 83, 715-726.
  3. Azzalini, A. and Capitanio, 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.
  4. Cabral, C. S., Lachos, V. H. and Prates, M. O. (2012). Multivariate mixture modeling using skew-normal independent distribution, Computational Statistics and Data Analysis, 56, 126-142.
  5. Cook, R. D. and Weisberg, S. (1994). An Introduction to Regression Graphics, 56, Wiley, New York.
  6. Lee, S. and McLachlan, G. J. (2011). On the fitting of mixtures of multivariate skew t-distributions via the EM algorithm, Technical Report of University of Queensland, Available from:
  7. Lin, T.-I. (2010). Robust mixture modeling using multivariate skew t distributions, Statistics and Computing, 20, 343-356.
  8. Lin, T.-I., Lee, J.-C. and Hsieh, W. J. (2007b). Robust mixture modeling using the skew t distributions, Statistics and Computing, 17, 81-92.
  9. Lin, T.-I., Lee, J.-C. and Yen, S. Y. (2007a). Finite mixture modeling using the skew normal distributions, Statistica Sinica, 17, 909-927.
  10. Liu, C. and Rubin, D. B. (1994). The ECME algorithm: a simple extension of EM and ECM with fast monotonic convergence, Biometroka, 81, 633-784.
  11. McLachlan, G. J. and Peel, D. (2000). Finite Mixture Models, Wiley, New York.
  12. Sahu, S. K., Dey, D. K. and Branco, M. D. (2003). A new class of multivariate skew distribution with application to Bayesian regression molel, The Canadian Journal of Statistics, 31, 129-150.

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  1. Diagnosis of Observations after Fit of Multivariate Skew t-Distribution: Identification of Outliers and Edge Observations from Asymmetric Data vol.25, pp.6, 2012,