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Singular and Non-singular Elliptically Contoured

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.
Stat., 1, 17-133.
Diaz-Garcia, J.A., Leiva, V., Galea, M. (2002) Singular elliptic distribution: density and applications.
Comm. Stat. Theor. Meth., 31, 665-681.
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
non-singular elliptic distributions. Comm. Stat. Theor. Meth., 32, 11-32.
Fang, K.T., Zhang, Y.T. (1990) Generalized Multivariate Analysis. Springer, Berlin.
Fang, K.T., Kotz, S., Zhang, Y.T. (1990) Symmetric Multivariate and Related Distributions. Chapman,
London.
Rao, C.R. (1965) Linear Statistical Inference and its Applications. Wiley, New York.
Riquelme, M., Leiva, V., Galea, M., Sanhueza, A. (2010) In°uence diagnostics on the coefficient of variation of elliptically contoured distributions. Journal of Applied Statistics (in press).
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.
 
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