A general class of multivariate skew-elliptical distributions (Q5949984)

Let \({\mathbf X}= (X_1,\dots, X_k)^T\) be a random vector elliptically distributed with location vector \(\mu\in \mathbb{R}^k\) and \(k\times k\) (positive definite) dispersion matrix \(\Sigma\), and assume that the vector \((X_0, X_1,\dots, X_k)^T\) is also elliptically distributed with location vector \((0,\mu)\) and dispersion matrix \(\left( \begin{smallmatrix} 1 & \delta^T \\ \delta & \Sigma \end{smallmatrix} \right)\) for some \(\delta= (\delta_1,\dots, \delta_k)^T\). Then the random vector NEWLINE\[NEWLINE{\mathbf Y}= \begin{cases} {\mathbf X}- \mu\quad\text{if} X_0>0 \\ -{\mathbf X}+\mu\quad\text{if} X_0 \leq 0\end{cases}NEWLINE\]NEWLINE is said to have a skew-elliptical distribution. The authors give several examples of skew-elliptical distributions, and compute moment generating functions, mean vectors and covariance matrices.