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On mixtures of skew normal and skew \(t\)-distributions

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Abstract

Finite mixtures of skew distributions have emerged as an effective tool in modelling heterogeneous data with asymmetric features. With various proposals appearing rapidly in the recent years, which are similar but not identical, the connection between them and their relative performance becomes rather unclear. This paper aims to provide a concise overview of these developments by presenting a systematic classification of the existing skew symmetric distributions into four types, thereby clarifying their close relationships. This also aids in understanding the link between some of the proposed expectation-maximization based algorithms for the computation of the maximum likelihood estimates of the parameters of the models. The final part of this paper presents an illustration of the performance of these mixture models in clustering a real dataset, relative to other non-elliptically contoured clustering methods and associated algorithms for their implementation.

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References

  • Arellano-Valle RB, Azzalini A (2006) On the unification of families of skew-normal distributions. Scand J Stat 33:561–574

    Article  MathSciNet  MATH  Google Scholar 

  • Arellano-Valle RB, Genton MG (2005) On fundamental skew distributions. J Multivar Anal 96:93–116

    Article  MathSciNet  MATH  Google Scholar 

  • Arellano-Valle RB, Genton MG (2010) Multivariate extended skew-\(t\) distributions and related families. METRON 68:201–234

    Article  MathSciNet  Google Scholar 

  • Arellano-Valle RB, Branco MD, Genton MG (2006) A unified view on skewed distributions arising from selections. Can J Stat 34:581–601

    Google Scholar 

  • Arellano-Valle RB, Castro LM, Genton MG, Gómez HW (2008) Bayesian inference for shape mixtures of skewed distributions, with application to regression analysis. Bayesian Anal 3:513–540

    Google Scholar 

  • Arnold BC, Beaver RJ, Meeker WQ (1993) The nontruncated marginal of a truncated bivariate normal distribution. Psychometrika 58:471–478

    Article  MathSciNet  MATH  Google Scholar 

  • Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12:171–178

    MathSciNet  MATH  Google Scholar 

  • Azzalini A (2005) The skew-normal distribution and related multivariate families. Scand J Stat 32:159–188

    Article  MathSciNet  MATH  Google Scholar 

  • Azzalini A, Capitanio A (1999) Statistical applications of the multivariate skew-normal distribution. J R Stat Soc Ser B 61(3):579–602

    Article  MathSciNet  MATH  Google Scholar 

  • Azzalini A, Capitanio A (2003) Distribution generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J R Stat Soc Ser B 65(2):367–389

    Article  MathSciNet  MATH  Google Scholar 

  • Azzalini A, Dalla Valle A (1996) The multivariate skew-normal distribution. Biometrika 83(4):715–726

    Article  MathSciNet  MATH  Google Scholar 

  • Basso RM, Lachos VH, Cabral CRB, Ghosh P (2010) Robust mixture modeling based on scale mixtures of skew-normal distributions. Comput Stat Data Anal 54:2926–2941

    Article  MathSciNet  Google Scholar 

  • Branco MD, Dey DK (2001) A general class of multivariate skew-elliptical distributions. J Multivar Anal 79:99–113

    Article  MathSciNet  MATH  Google Scholar 

  • Cabral CRB, Lachos VH, Prates MO (2012) Multivariate mixture modeling using skew-normal independent distributions. Comput Stat Data Anal 56:126–142

    Article  MathSciNet  MATH  Google Scholar 

  • Contreras-Reyes JE, Arellano-Valle RB (2012) Growth curve based on scale mixtures of skew-normal distributions to model the age-length relationship of cardinalfish (epigonus crassicaudus). arXiv:12125180 [statAP]

  • Franczak BC, Browne RP, McNicholas PD (2012) Mixtures of shifted asymmetric laplace distributions. arXiv:12071727 [statME]

  • Frühwirth-Schnatter S, Pyne S (2010) Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-\(t\) distributions. Biostatistics 11:317–336

    Article  Google Scholar 

  • Genton MG (ed) (2004) Skew-elliptical Distributions and their Applications: a Journey beyond Normality. Chapman & Hall/CRC, Boca Raton/Florida

  • Genton MG, Loperfido N (2005) Generalized skew-elliptical distributions and their quadratic forms. Ann Inst Stat Math 57:389–401

    Article  MathSciNet  Google Scholar 

  • González-Farás G, Domínguez-Molinz JA, Gupta AK (2004) Additive properties of skew normal random vectors. J Stat Plan Inference 126:521–534

    Article  Google Scholar 

  • Gupta AK (2003) Multivariate skew-\(t\) distribution. Statistics 37:359–363

    Article  MathSciNet  MATH  Google Scholar 

  • Gupta AK, González-Faríaz G, Domínguez-Molina JA (2004) A multivariate skew normal distribution. J Multivar Anal 89:181–190

    Google Scholar 

  • Ho HJ, Lin TI, Chen HY, Wang WL (2012) Some results on the truncated multivariate \(t\) distribution. J Stat Plan Inference 142:25–40

    Article  MathSciNet  MATH  Google Scholar 

  • Iversen DH (2010) Closed-skew distributions: simulation, inversion and parameter estimation. Norwegian University of Science and Technology, Master’s thesis

  • Karlis D, Santourian A (2009) Model-based clustering with non-elliptically contoured distributions. Stat Comput 19:73–83

    Article  MathSciNet  Google Scholar 

  • Lachos VH, Ghosh P, Arellano-Valle RB (2010) Likelihood based inference for skew normal independent linear mixed models. Stat Sin 20:303–322

    MathSciNet  MATH  Google Scholar 

  • Lee SX, McLachlan GJ (2011) On the fitting of mixtures of multivariate skew t-distributions via the EM algorithm. arXiv:11094706 [statME]

  • Lee SX, McLachlan GJ (2013) Finite mixtures of multivariate skew \(t\)-distributions: some recent and new results. Stat Comput

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

    Article  MATH  Google Scholar 

  • Lin TI (2010) Robust mixture modeling using multivariate skew \(t\) distribution. Stat Comput 20:343–356

    Article  MathSciNet  Google Scholar 

  • Lin TI, Lee JC, Hsieh WJ (2007a) Robust mixture modeling using the skew-\(t\) distribution. Stat Comput 17:81–92

    Article  MathSciNet  Google Scholar 

  • Lin TI, Lee JC, Yen SY (2007b) Finite mixture modelling using the skew normal distribution. Stat Sin 17:909–927

    MathSciNet  MATH  Google Scholar 

  • Lin TI, Ho HJ, Lee CR (2013) Flexible mixture modelling using the multivariate skew-\(t\)-normal distribution. Stat Comput. doi:10.1007/s11222-013-9386-4

  • Liseo B, Loperfido N (2003) A Bayesian interpretation of the multivariate skew-normal distribution. Stat Probab Lett 61:395–401

    Article  MathSciNet  MATH  Google Scholar 

  • Ma Y, Genton MG (2004) A flexible class of skew-symmetric distributions. Scand J Stat 31:459–468

    Google Scholar 

  • Prates M, Lachos V, Cabral C (2011) mixsmsn: fitting finite mixture of scale mixture of skew-normal distributions. http://CRAN.R-project.org/package=mixsmsn, R package version 1.0-7

  • Pyne S, Hu X, Wang K, Rossin E, Lin TI, Maier LM, Baecher-Allan C, McLachlan GJ, Tamayo P, Hafler DA, De Jager PL, Mesirow JP (2009) Automated high-dimensional flow cytometric data analysis. Proc Natl Acad Sci USA 106:8519–8524

    Article  Google Scholar 

  • Riggi S, Ingrassia S (2013) Modeling high energy cosmic rays mass composition data via mixtures of multivariate skew-\(t\) distributions. arXiv:13011178 [astro-phHE]

  • Sahu SK, Dey DK, Branco MD (2003) A new class of multivariate skew distributions with applications to Bayesian regression models. Can J Stat 31:129–150

    Article  MathSciNet  MATH  Google Scholar 

  • Soltyk S, Gupta R (2011) Application of the multivariate skew normal mixture model with the EM algorithm to value-at-risk. MODSIM 2011—19th international congress on modelling and simulation, Perth

  • Vrbik I, McNicholas PD (2012) Analytic calculations for the EM algorithm for multivariate skew \(t\)-mixture models. Stat Probab Lett 82:1169–1174

    Article  MathSciNet  MATH  Google Scholar 

  • Vrbik I, McNicholas PD (2013) Parsimonious skew mixture models for model-based clustering and classification. arXiv:13022373 [statCO]

  • Wang K, McLachlan GJ, Ng SK, Peel D (2009) EMMIX-skew: EM algorithm for mixture of multivariate skew Normal/\(t\) distributions. http://www.maths.uq.edu.au/gjm/mix_soft/EMMIX-skew, R package version 1.0-12

  • Wang K, Ng SK, McLachlan GJ (2009) Multivariate skew \(t\) mixture models: applications to fluorescence-activated cell sorting data. In: Shi H, Zhang Y, Bottema MJ, Lovell BC, Maeder AJ (eds) DICTA 2009 (conference of digital image computing: techniques and applications, Melbourne). IEEE Computer Society, Los Alamitos, pp 526–531

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Acknowledgments

This work is supported by a grant from the Australian Research Council. The authors would like to thank the Editor and reviewers for their very helpful comments and suggestions.

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Authors and Affiliations

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

    Sharon X. Lee & Geoffrey J. McLachlan

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  1. Sharon X. Lee
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  2. Geoffrey J. McLachlan
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Correspondence to Geoffrey J. McLachlan.

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Lee, S.X., McLachlan, G.J. On mixtures of skew normal and skew \(t\)-distributions. Adv Data Anal Classif 7, 241–266 (2013). https://doi.org/10.1007/s11634-013-0132-8

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  • DOI: https://doi.org/10.1007/s11634-013-0132-8

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