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A study of generalized skew-normal distribution. (English) Zbl 1440.62197

Summary: Following the paper by M. G. Genton and N. M. R. Loperfido [Ann. Inst. Stat. Math. 57, No. 2, 369–401 (2005; Zbl 1083.62043)], we say that \(\mathbf{Z}\) has a generalized skew-normal distribution, if its probability density function (p.d.f.) is given by \(f(\mathbf{z}) = 2\varphi_p(\mathbf{z}; \boldsymbol{\xi},\boldsymbol{\Omega})\pi (\mathbf{z}-\boldsymbol{\xi})\), \(\mathbf{z} \in\mathbb{R}^p\), where \(\varphi_p(\cdot;\boldsymbol{\xi},\boldsymbol{\Omega})\) is the \(p\)-dimensional normal p.d.f. with location vector \(\boldsymbol{\xi}\) and scale matrix \(\boldsymbol{\Omega}\), \(\boldsymbol{\xi} \in \mathbb{R}^p\), \(\boldsymbol{\Omega}>0\), and \(\pi\) is a skewing function from \(\mathbb{R}^p\) to \(\mathbb{R}\), that is \(0 \leq \pi (\mathbf{z}) \leq 1\) and \(\pi(-\mathbf{z}) = 1-\pi(\mathbf{z})\), \(\forall \mathbf{z} \in\mathbb{R}^p\). First the distribution of linear transformations of \(\mathbf{Z}\) are studied, and some moments of \(\mathbf{Z}\) and its quadratic forms are derived. Next we obtain the joint moment-generating functions (m.g.f.’s) of linear and quadratic forms of \(\mathbf{Z}\) and then investigate conditions for their independence. Finally explicit forms for the above distributions, m.g.f.’s and moments are derived when \(\pi(\mathbf{z}) = \kappa(\boldsymbol{\alpha}'\mathbf{z})\), where \(\boldsymbol{\alpha} \in \mathbb{R}^p\) and \(\kappa\) is the normal, Laplace, logistic or uniform distribution function.

MSC:

62H10 Multivariate distribution of statistics
62H05 Characterization and structure theory for multivariate probability distributions; copulas
60E05 Probability distributions: general theory

Citations:

Zbl 1083.62043
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References:

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