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A general class of multivariate skew-elliptical distributions. (English) Zbl 0992.62047
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 ${\mathbf Y}= \begin{cases} {\mathbf X}- \mu\quad\text{if} X_0>0 \\ -{\mathbf X}+\mu\quad\text{if} X_0 \leq 0\end{cases}$ 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.

##### MSC:
 62H10 Multivariate distribution of statistics 60E05 Probability distributions: general theory
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##### References:
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