Summary
A class of multivariate extended skew-t (EST) distributions is introduced and studied in detail, along with closely related families such as the subclass of extended skew-normal distributions. Besides mathematical tractability and modeling flexibility in terms of both skewness and heavier tails than the normal distribution, the most relevant properties of the EST distribution include closure under conditioning and ability to model lighter tails as well. The first part of the present paper examines probabilistic properties of the EST distribution, such as various stochastic representations, marginal and conditional distributions, linear transformations, moments and in particular Mardia’s measures of multivariate skewness and kurtosis. The second part of the paper studies statistical properties of the EST distribution, such as likelihood inference, behavior of the profile log-likelihood, the score vector and the Fisher information matrix. Especially, unlike the extended skew-normal distribution, the Fisher information matrix of the univariate EST distribution is shown to be non-singular when the skewness is set to zero. Finally, a numerical application of the conditional EST distribution is presented in the context of confidential data perturbation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adcock, C. J. (2010) Asset pricing and portfolio selection based on the multivariate extended skew-Student-t distribution, Annals of Operations Research, 176, 221–234.
Arellano-Valle, R.B. (2010) On the information matrix of the multivariate skew-t model, Metron, 68, 371–386.
Arellano-Valle, R.B. and Azzalini, A. (2006) On the unification of families of skew-normal distributions, Scandina vian Journal of Statistics, 33, 561–574.
Arellano-Valle, R.B., Branco, M. D. and Genton, M. G. (2006) A unified view on skewed distributions arising from selections, Canadian Journal of Statistics, 34, 581–601.
Arellano-Valle, R.B., del Pino, G. and San Martín, E. (2002) Definition and probabilistic properties of skew distributions, Statistics and Probability Letters, 58, 111–121.
Arellano-Valle, R.B. and Genton, M. G. (2005) On fundamental skew distributions, Journal of Multivariate Analysis, 96, 93–116.
Arnold, B. C. and Beaver, R. J.(2000a) The skew-Cauchy distribution, Statistics and Probability Letters, 49, 285–290.
Arnold, B. C. and Beaver, R. J.(2000b) Hidden truncation models, Sankhyā Ser. A, 62, 22–35.
Azzalini, A. (1985) A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12, 171–178.
Azzalini, A. (2005) The skew-normal distribution and related multivariate families (with discussion by Marc G. Genton and a rejoinder by the author), Scandinavian Journal of Statistics, 32, 159–200.
Azzalini, A. (2006) R package sn: The skew-normal and skew-t distributions (version 0.4–2), URL http://azzalini.stat.unipd.it/SN.
Azzalini, A. and Capitanio, A. (1999) Statistical applications of the multivariate skew-normal distribution, Journal of the Royal Statistical Society Series B, 61, 579–602.
Azzalini, A. and Capitanio, A. (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution, Journal of the Royal Statistical Society Series B, 65, 367–389.
Azzalini, A. and Dalla Valle, A. (1996) The multivariate skew-normal distribution, Biometrika, 83, 715–726.
Azzalini, A. and Genton, M. G. (2008) Robust likelihood methods based on the skew-t and related distributions, International Statistical Review, 76, 106–129.
Branco, M. D. and Dey, D. K. (2001) A general class of multivariate skew-elliptical distributions, Journal of Multivariate Analysis, 79, 99–113.
Capitanio, A., Azzalini, A. and Stanghellini, E. (2003) Graphical models for skew-normal variates, Scandinavian Journal of Statistics, 30, 129–144.
Cook, R. D. and Weisberg, S. (1994) An Introduction to Regression Graphics, Wiley, New York.
Diciccio, T. and Monti, A. (2009) Inferential aspects of the skew-t distribution, Manuscript.
Fang, K.-T., Kotz, S. and Ng, K.-W. (1990) Symmetric Multivariate and Related Distributions, Monographs on Statistics and Applied Probability, 36, Chapman and Hall, Ltd., London.
Genton, M. G. (2004) Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality, Edited Volume, Chapman & Hall/CRC, Boca Raton, FL, pp. 416.
Genz, A. (1992) Numerical computation of multivariate Normal probabilities, Journal of Computational and Graphical Statistics, 1, 141–149.
Genz, A. and Bretz, F. (2002) Methods for the computation of multivariate t-probabilities, Journal of Computational and Graphical Statistics, 11, 950–971.
González-Farías, G., Domínguez-Molina, J. A. and Gupta, A. K. (2004) The closed skew-normal distribution, In: Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality, M. G. Genton (ed.), Chapman & Hall / CRC, Boca Raton, FL, 25–42.
Lee, S., Genton, M. G. and Arellano-Valle, R. B. (2010) Perturbation of numerical confidential data via skew-t distributions, Management Science, 56, 318–333.
Mardia, K. V. (1970) Measures of multivariate skewness and kurtosis with applications, Biometrika, 36, 519–530.
Muralidhar, K. and Sarathy, R. (2003) A theoretical basis for perturbation methods, Statistics and Computing, 13, 329–335.
Park, J. W., Genton, M. G. and Ghosh, S. K. (2007) Censored time series analysis with autoregressive moving average models, Canadian Journal of Statistics, 35, 151–168.
Pewsey, A. (2006) Some observations on a simple means of generating skew distributions, In: Advances in Distribution Theory, Order Statistics, and Inference, N. Balakrishnan, E. Castillo and J. M. Sarabia (eds.), Boston: Statistics for Industry and Technology, Birkhäuser 75–84.
Stuart, A. and Ord, J. K. (1987) Kendall’s Advanced Theory of Statistics, Vol. 1: Distribution Theory, Oxford University Press, Fifth Edition, New York.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Arellano-Valle, R.B., Genton, M.G. Multivariate extended skew-t distributions and related families. METRON 68, 201–234 (2010). https://doi.org/10.1007/BF03263536
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03263536