A special and interesting multivariate class that contains the normal model, as well as non-normal models, is the family of elliptically contoured (EC) distributions. Numerous authors have studied the EC models; see, for example, the books by Fang and Zhang (1990) and Fang et al. (1990), and the articles by Diaz-Garcia et al. (2003), Savalli et al. (2006), and Riquelme et al. (2010) for more recent results. In this talk, singular and non-singular EC distributions are presented. An explicit expression for the density of an n-dimensional random vector with a singular EC distribution is discussed; see Rao (1965), Diaz-Garcia et al. (2002), and Arellano-Valle and Genton (2010). Based on the singular and non-singular cases, the generalized central, non-central and doubly non-central X^2, t and F distributions are analyzed; see Diaz-Garcia and Leiva (2003). Some particular cases of these classes of distributions are considered. Finally, the results are applied to the study of the distribution of the residuals of an EC linear model, the distribution of the t-statistic based on a sample from an EC population, and the inference on the cofficient of variation of this type of populations, among other applications.
References
Arellano-Valle, R.B., Genton, M.G. (2010) Multivariate uni¯ed skew-elliptical distributions. Chilean J.
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Diaz-Garcia, J.A., Leiva, V., Galea, M. (2002) Singular elliptic distribution: density and applications.
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Diaz-Garcia, J.A., Galea, M., Leiva, V. (2003) In°uence diagnostics for multivariate elliptic regression linear models. Comm. Stat. Theor. Meth., 32, 625-641.
Diaz-Garcia, J.A., Leiva, V. (2003) Doubly non-central t and F distribution obtained under singular and
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Fang, K.T., Zhang, Y.T. (1990) Generalized Multivariate Analysis. Springer, Berlin.
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Riquelme, M., Leiva, V., Galea, M., Sanhueza, A. (2010) In°uence
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http://dx.doi.org/10.1080/02664760903521427
Savalli, C., Paula, G.A., Cysneiros, F.J.A. (2006) Assessment of variance components in elliptical linear mixed models. Stat. Model., 6, 59-76. |