×

On mixtures of skew normal and skew \(t\)-distributions. (English) Zbl 1273.62115

Summary: 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 data set, relative to other non-elliptically contoured clustering methods and associated algorithms for their implementation.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62H12 Estimation in multivariate analysis
65C60 Computational problems in statistics (MSC2010)

Software:

mixsmsn; sn; EMMIX-skew
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Arellano-Valle RB, Azzalini A (2006) On the unification of families of skew-normal distributions. Scand J Stat 33:561–574 · Zbl 1117.62051
[2] Arellano-Valle RB, Genton MG (2005) On fundamental skew distributions. J Multivar Anal 96:93–116 · Zbl 1073.62049
[3] Arellano-Valle RB, Genton MG (2010) Multivariate extended skew- $$t$$ distributions and related families. METRON 68:201–234 · Zbl 1301.62016
[4] Arellano-Valle RB, Branco MD, Genton MG (2006) A unified view on skewed distributions arising from selections. Can J Stat 34:581–601 · Zbl 1121.60009
[5] 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
[6] Arnold BC, Beaver RJ, Meeker WQ (1993) The nontruncated marginal of a truncated bivariate normal distribution. Psychometrika 58:471–478 · Zbl 0794.62075
[7] Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12:171–178 · Zbl 0581.62014
[8] Azzalini A (2005) The skew-normal distribution and related multivariate families. Scand J Stat 32:159–188 · Zbl 1091.62046
[9] Azzalini A, Capitanio A (1999) Statistical applications of the multivariate skew-normal distribution. J R Stat Soc Ser B 61(3):579–602 · Zbl 0924.62050
[10] 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 · Zbl 1065.62094
[11] Azzalini A, Dalla Valle A (1996) The multivariate skew-normal distribution. Biometrika 83(4):715–726 · Zbl 0885.62062
[12] 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 · Zbl 1284.62193
[13] Branco MD, Dey DK (2001) A general class of multivariate skew-elliptical distributions. J Multivar Anal 79:99–113 · Zbl 0992.62047
[14] Cabral CRB, Lachos VH, Prates MO (2012) Multivariate mixture modeling using skew-normal independent distributions. Comput Stat Data Anal 56:126–142 · Zbl 1239.62058
[15] 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]
[16] Franczak BC, Browne RP, McNicholas PD (2012) Mixtures of shifted asymmetric laplace distributions. arXiv:12071727 [statME]
[17] 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
[18] Genton MG (ed) (2004) Skew-elliptical Distributions and their Applications: a Journey beyond Normality. Chapman & Hall/CRC, Boca Raton/Florida · Zbl 1069.62045
[19] Genton MG, Loperfido N (2005) Generalized skew-elliptical distributions and their quadratic forms. Ann Inst Stat Math 57:389–401 · Zbl 1083.62043
[20] González-Farás G, Domínguez-Molinz JA, Gupta AK (2004) Additive properties of skew normal random vectors. J Stat Plan Inference 126:521–534 · Zbl 1076.62052
[21] Gupta AK (2003) Multivariate skew- $$t$$ distribution. Statistics 37:359–363 · Zbl 1037.62045
[22] Gupta AK, González-Faríaz G, Domínguez-Molina JA (2004) A multivariate skew normal distribution. J Multivar Anal 89:181–190 · Zbl 1036.62043
[23] Ho HJ, Lin TI, Chen HY, Wang WL (2012) Some results on the truncated multivariate $$t$$ distribution. J Stat Plan Inference 142:25–40 · Zbl 1229.62068
[24] Iversen DH (2010) Closed-skew distributions: simulation, inversion and parameter estimation. Norwegian University of Science and Technology, Master’s thesis
[25] Karlis D, Santourian A (2009) Model-based clustering with non-elliptically contoured distributions. Stat Comput 19:73–83
[26] Lachos VH, Ghosh P, Arellano-Valle RB (2010) Likelihood based inference for skew normal independent linear mixed models. Stat Sin 20:303–322 · Zbl 1186.62071
[27] Lee SX, McLachlan GJ (2011) On the fitting of mixtures of multivariate skew t-distributions via the EM algorithm. arXiv:11094706 [statME]
[28] Lee SX, McLachlan GJ (2013) Finite mixtures of multivariate skew $$t$$ -distributions: some recent and new results. Stat Comput
[29] Lin TI (2009) Maximum likelihood estimation for multivariate skew normal mixture models. J Multivar Anal 100:257–265 · Zbl 1152.62034
[30] Lin TI (2010) Robust mixture modeling using multivariate skew $$t$$ distribution. Stat Comput 20:343–356
[31] Lin TI, Lee JC, Hsieh WJ (2007a) Robust mixture modeling using the skew- $$t$$ distribution. Stat Comput 17:81–92
[32] Lin TI, Lee JC, Yen SY (2007b) Finite mixture modelling using the skew normal distribution. Stat Sin 17:909–927 · Zbl 1133.62012
[33] 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
[34] Liseo B, Loperfido N (2003) A Bayesian interpretation of the multivariate skew-normal distribution. Stat Probab Lett 61:395–401 · Zbl 1101.62342
[35] Ma Y, Genton MG (2004) A flexible class of skew-symmetric distributions. Scand J Stat 31:459–468 · Zbl 1063.62079
[36] 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
[37] 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
[38] Riggi S, Ingrassia S (2013) Modeling high energy cosmic rays mass composition data via mixtures of multivariate skew- $$t$$ distributions. arXiv:13011178 [astro-phHE]
[39] 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 · Zbl 1039.62047
[40] 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
[41] Vrbik I, McNicholas PD (2012) Analytic calculations for the EM algorithm for multivariate skew $$t$$ -mixture models. Stat Probab Lett 82:1169–1174 · Zbl 1244.65012
[42] Vrbik I, McNicholas PD (2013) Parsimonious skew mixture models for model-based clustering and classification. arXiv:13022373 [statCO]
[43] 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
[44] 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.