Family of multivariate generalized \(t\) distributions
A random vector (r.v.) is said to have a \(p\)--dimensional multivariate generalized \(t\) distribution, denoted by \(MGT_p(\mu,\Sigma,\lambda,\beta,q)\), if its density is given by \[ f(x;\mu,\Sigma,\lambda,\beta,q)=C\lambda^{-p/2}| \Sigma| ^{-1/2} [q+(s/\lambda)^{\beta}]^{-(q+p/2\beta)}, \] where \(\Sigma\) is a \(p\times p\) positive--definite symmetric matrix, \(\lambda,\beta,q>0\), \(\mu\in\mathbf R^p\), and the normalizing constant is given in terms of gamma functions. These distributions belong to the family of elliptically contoured distributions and generalize the GT distributions introduced by \textit{J.B. McDonald} and \textit{W.K. Newey} [Econ. Theory 16, No. 4, 1049-1074 (1988)]. Characteristic functions, average, variance, multivariate asymmetry and kurtosis coefficients are given as well as the connection with a r.v. having a uniform distribution on the unit sphere of \(\mathbf R ^p\). Finally, the distribution of the r.v. \(Y=CX+b\), where \(X\sim MGT_p(\mu,\Sigma,\lambda,\beta,q)\), is studied.
scientific article; zbMATH DE number 2091278