Finite Mixtures of Matrix Variate t Distributions

Abstract

Finite mixture of multivariate t distributions (Peel and McLachlan, 2000) was introduced as an alternative to the finite mixture of multivariate normal distributions to model datasets with heavy tails. In this study, we define the finite mixture of matrix variate t distributions as an extension of finite mixture of multivariate t distributions. Mixture of matrix variate t distributions can provide an alternative robust model to the mixture of matrix variate normal distributions (Viroli, 2011) for modeling matrix variate datasets with heavy tails. We give an Expectation Maximization (EM) algorithm to find the maximum likelihood (ML) estimators for the parameters of interest. We also provide a small simulation study to illustrate the performance of the proposed EM algorithm for finding estimates.

Keywords

• Finite mixtures
• matrix variate t
• matrix variate normal
• ML estimator
• EM algorithm

References

1. Bulut, Y. M., Arslan, O. Matrix variate t-distribution: properties, parameter estimation and application to robust statistical analysis, (submitted), (2015).
2. Cabral, C., Lachos, V., Prates, M. Multivariate mixture modeling using skew-normal independent distributions. Computational Statistics and Data Analysis, 56, 126–142, (2012).
3. Dawid, A.P. Some matrix-variate distribution theory: notational considerations and a Bayesian application. Biometrika 68, 265–274, (1981).
4. Dempster, A. P., Laird, N. M., Rubin, D. B. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39(1), 1-38, (1977).
5. DeWall, D. J. Matrix-variate distributions. In: Knotz, S., Johnson, N.L. (eds.): Encyclopedia of Statistical Sciences, vol. 5, pp. 326–333, Wiley, New York, (1988).
6. Dutilleul, P. The MLE algorithm for the matrix normal distribution. Journal of Statistical Computation and Simulation, 64 (2), 105–123, (1999).
7. Frühwirth-Schnatter, S. Finite Mixture and Markov Switching Models. Springer, New York, (2006).
8. Gupta, A. K., Varga, T., Bodnar, T. Elliptically Contoured Models in Statistics and Portfolio Theory. Springer, New York, (2013).

9. Lin, T. I. Maximum likelihood estimation for multivariate skew normal mixture models. Journal of Multivariate Analysis, 100, 257–265, (2009).
10. Lin, T. I. Robust mixture modeling using multivariate skew t distributions. Statistics and Computing, 20, 343–356, (2010).
11. McLachlan, G. J., Basord, K. E. Mixture Models: Inference and Application to Clustering. Marcel Dekker, New York, (1988).
12. McLachlan, G. J., Peel, D. Finite Mixture Models. Wiley, New York, (2000).
13. O’Hagan, A., Murphy, T. B., Gormley, I. C., McNicholas, P., Karlis, D. Clustering with the multivariate normal inverse Gaussian distribution. Computational Statistics and Data Analysis (in press), (2015).
14. Peel, D., McLachlan, G. J. Robust mixture modelling using the t distribution. Statistics and Computing, 10, 339-348, (2000).
15. Pyne, S., Hu, X., Wang, K., Rossin, E., Lin, T., Maier, L. M. Automated high-dimensional flow cytometric data analysis. Proceedings of the National Academy of Sciences USA 106, 8519-8524, (2009).
16. Rowe, B. R. Multivariate Bayesian Statistics. Chapman and Hall/CRC, London/Boca Raton, (2003).
17. Srivastava, M., Khatri, C. An Introduction to Multivariate Statistics. North Holland, New York, USA, (1979).
18. Srivastava, M., Nahtman, T., von Rosen, D. Models with a Kronecker product covariance structure. Estimation and testing. Mathematical Methods of Statistics, 17(4), 357–370, (2008).
19. Titterington, D. M., Smith, A. F. M., Markov, U. E. Statistical Analysis of Finite Mixture Distributions. Wiley, New York, (1985).
20. Viroli, C. Finite mixtures of matrix normal distributions for classifying three-way data. Statistical Computing, 21, 511–522, (2011).