Maximum likelihood estimation for multivariate skew normal mixture models. (English) Zbl 1152.62034

Summary: This paper provides a flexible mixture modeling framework using the multivariate skew normal distribution. A feasible EM algorithm is developed for finding the maximum likelihood estimates of the parameters in this context. A general information-based method for obtaining the asymptotic covariance matrix of the maximum likelihood estimators is also presented. The proposed methodology is illustrated with a real example and results are also compared with those obtained from fitting normal mixtures.


62H12 Estimation in multivariate analysis
62H10 Multivariate distribution of statistics
65C60 Computational problems in statistics (MSC2010)
62F10 Point estimation
62B10 Statistical aspects of information-theoretic topics


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[1] Arellano-Valle, R.B.; Bolfarine, H.; Lachos, V.H., Bayesian inference for skew-normal linear mixed models, J. appl. stat., 34, 663-682, (2007)
[2] Arellano-Valle, R.B.; Genton, M.G., On fundamental skew distributions, J. multivariate anal., 96, 93-116, (2005) · Zbl 1073.62049
[3] Azzalini, A., A class of distributions which includes the normal ones, Scand. J. statist., 12, 171-178, (1985) · Zbl 0581.62014
[4] Azzalini, A.; Capitaino, A., Distributions generated by perturbation of symmetry with emphasis on a multivariate skew \(t\)-distribution, Roy. statist. soc. ser. B, 65, 367-389, (2003) · Zbl 1065.62094
[5] Azzalini, A.; Dalla Valle, A., The multivariate skew-normal distribution, Biometrika, 83, 715-726, (1996) · Zbl 0885.62062
[6] Basford, K.E.; Greenway, D.R.; McLachlan, G.J.; Peel, D., Standard errors of fitted means under normal mixture, Comp. statist., 12, 1-17, (1997) · Zbl 0924.62055
[7] Dempster, A.P.; Laird, N.M.; Rubin, D.B., Maximum likelihood from incomplete data via the EM algorithm (with discussion), J. roy. statist. soc. ser. B, 39, 1-38, (1977) · Zbl 0364.62022
[8] Efron, B.; Tibshirani, R., Bootstrap method for standard errors, confidence intervals, and other measures of statistical accuracy, Statist. sci., 1, 54-77, (1986) · Zbl 0587.62082
[9] Flury, B.; Riedwyl, H., Multivariate statistics, a practical approach, (1988), Cambridge University Press Cambridge
[10] Frühwirth-Schnatter, S., Finite mixture and Markov switching models, (2006), Springer New York · Zbl 1108.62002
[11] Hartigan, J.A.; Wong, M.A., Algorithm AS 136: A \(K\)-means clustering algorithm, Appl. stat., 28, 100-108, (1979) · Zbl 0447.62062
[12] Genz, A., Numerical computation of multivariate normal probabilities, J. comput. graph. statist., 1, 141-150, (1992)
[13] Genz, A., Comparison of methods for the computation of multivariate normal probabilities, Comp. sci. statist., 25, 400-405, (1993)
[14] Gupta, A.F.; González-Farías, G.; Domínguez-Monila, J.A., A multivariate skew normal distribution, J. multivariate anal., 89, 181-190, (2004) · Zbl 1036.62043
[15] Styan, G.P.H., Hadamard products and multivariate statistical analysis, Linear algebra appl., 6, 217-240, (1973) · Zbl 0255.15002
[16] Lin, T.I.; Lee, J.C.; Hsieh, W.J., Robust mixture modeling using the skew \(t\) distribution, Statist. comp., 17, 81-92, (2007)
[17] Lin, T.I.; Lee, J.C.; Yen, S.Y., Finite mixture modelling using the skew normal distribution, Statist. sinica, 17, 909-927, (2007) · Zbl 1133.62012
[18] Ma, Y.; Genton, M.G., Flexible class of skew-symmetric distribtions, Scand. J. statist., 31, 459-468, (2004) · Zbl 1063.62079
[19] McLachlan, G.J.; Basord, K.E., Mixture models: inference and application to clustering, (1988), Marcel Dekker New York
[20] McLachlan, G.J.; Peel, D., Finite mixture models, (2000), Wiely New York · Zbl 0963.62061
[21] Redner, R.A.; Walker, H.F., Mixture densities, maximum likelihood and the EM algorithm, SIAM rev., 26, 195-239, (1984) · Zbl 0536.62021
[22] Sahu, S.K.; Dey, D.K.; Branco, M.D., A new class of multivariate skew distributions with application to Bayesian regression models, Canad. J. statist., 31, 129-150, (2003) · Zbl 1039.62047
[23] Tallis, G.M., The moment generating function of the truncated multi-normal distribution, J. roy. statist. soc. ser. B, 23, 223-229, (1961) · Zbl 0107.14206
[24] Titterington, D.M.; Smith, A.F.M.; Markov, U.E., Statistical analysis of finite mixture distributions, (1985), Wiely New York · Zbl 0646.62013
[25] Wu, C.F.J., On the convergence properties of the EM algorithm, Ann. statist., 11, 95-103, (1983) · Zbl 0517.62035
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