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Finite mixtures of canonical fundamental skew \(t\)-distributions

The unification of the restricted and unrestricted skew \(t\)-mixture models

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Abstract

This paper introduces a finite mixture of canonical fundamental skew \(t\) (CFUST) distributions for a model-based approach to clustering where the clusters are asymmetric and possibly long-tailed (in: Lee and McLachlan, arXiv:1401.8182 [statME], 2014b). The family of CFUST distributions includes the restricted multivariate skew \(t\) and unrestricted multivariate skew \(t\) distributions as special cases. In recent years, a few versions of the multivariate skew \(t\) (MST) mixture model have been put forward, together with various EM-type algorithms for parameter estimation. These formulations adopted either a restricted or unrestricted characterization for their MST densities. In this paper, we examine a natural generalization of these developments, employing the CFUST distribution as the parametric family for the component distributions, and point out that the restricted and unrestricted characterizations can be unified under this general formulation. We show that an exact implementation of the EM algorithm can be achieved for the CFUST distribution and mixtures of this distribution, and present some new analytical results for a conditional expectation involved in the E-step.

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Acknowledgments

We would like to thank Professor Seung-Gu Kim for helpful comments on this topic. The work of the authors was supported by an Australian Research Council Discovery Grant.

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  1. Department of Mathematics, University of Queensland, St. Lucia, 4072, Australia

    Sharon X. Lee & Geoffrey J. McLachlan

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  1. Sharon X. Lee
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Lee, S.X., McLachlan, G.J. Finite mixtures of canonical fundamental skew \(t\)-distributions. Stat Comput 26, 573–589 (2016). https://doi.org/10.1007/s11222-015-9545-x

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