Abstract
In this paper we introduce the concept of weak homoscedasticity for covariance matrices of the component densities, in the framework of constrained formulations of the maximum likelihood estimation for mixture models. Further, we give a test for assessing weak homoscedasticity in two sample data. Based on such approach, we present how to implement a constrained EM algorithm for mixtures of t-distributions. The proposal is illustrated on the ground of numerical experiments which show its usefulness in data modeling and classification.
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References
Banfield, J. D., & Raftery, A. E. (1993). Model-based Gaussian and non Gaussian clustering. Biometrics, 49, 803–821.
Fang, K. T., & Anderson, T. W. (1990). Statistical inference in elliptically contoured and related distributions. New York: Alberton.
Fraley, C., & Raftery, A. E. (2002). Model-based clustering, discriminant analysis, and density estimation. Journal of the American Statistical Association, 97, 611–631.
Greselin, F., & Ingrassia, S. (2009). Constrained monotone EM algorithms for mixtures of multivariate t-distributions. Statistics and Computing. doi: 10.1007/s11222-008-9112-9.
Hathaway, R. J. (1986). A constrained formulation of maximum-likelihood estimation for normal mixture distributions. The Annals of Statistics, 13, 795–800.
Hawkins, D. M. (1981). A new test for multivariate normality and homoscedasticity. Technometrics, 23, 105–110.
Hennig, C. (2004). Breakdown points for maximum likelihood estimators of location-scale mixtures. The Annals of Statistics, 32, 1313–1340.
Ingrassia, S. (2004). A likelihood-based constrained algorithm for multivariate normal mixture models. Statistical Methods and Applications, 13, 151–166.
Ingrassia, S., & Rocci, R. (2007). Constrained monotone EM algorithms for finite mixture of multivariate Gaussians. Computational Statistics and Data Analysis, 51, 5339–5351.
Kotz, S., & Nadarajah, S. (2004). Multivariate t distributions and their applications. New York: Cambridge University Press.
Lin, T. I., Lee, J. C., & Hsieh, W. J. (2007). Robust mixture modeling using the skew t distribution. Statistics and computing, 17, 81–92.
McLachlan, G. J., & Peel, D. (2000). Finite mixture models. New York: Wiley.
Murtagh, F., & Raftery, A. E. (1984). Fitting straight lines to point patterns. Pattern Recognition, 17, 479–483.
Peel, D., & McLachlan, G. J. (2000). Robust mixture modelling using the t distribution. Statistics and Computing, 10, 339–348.
Acknowledgements
The authors thank the referees for their interesting comments and suggestions which contributed to improving an earlier version of the paper.
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Greselin, F., Ingrassia, S. (2009). Weakly Homoscedastic Constraints for Mixtures of t-Distributions. In: Fink, A., Lausen, B., Seidel, W., Ultsch, A. (eds) Advances in Data Analysis, Data Handling and Business Intelligence. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01044-6_20
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DOI: https://doi.org/10.1007/978-3-642-01044-6_20
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