A study of generalized skew-normal distribution |
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Authors: | Wen-Jang Huang Arjun K Gupta |
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Institution: | 1. Department of Applied Mathematics , National University of Kaohsiung , Kaohsiung , 81148 , Taiwan, Republic of China;2. Department of Mathematics and Statistics , Bowling Green State University , Bowling Green , OH , 43403 , USA |
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Abstract: | Following the paper by Genton and Loperfido Generalized skew-elliptical distributions and their quadratic forms, Ann. Inst. Statist. Math. 57 (2005), pp. 389–401], we say that Z has a generalized skew-normal distribution, if its probability density function (p.d.f.) is given by f(z)=2φ p (z; ξ, Ω)π (z?ξ), z∈? p , where φ p (·; ξ, Ω) is the p-dimensional normal p.d.f. with location vector ξ and scale matrix Ω, ξ∈? p , Ω>0, and π is a skewing function from ? p to ?, that is 0≤π (z)≤1 and π (?z)=1?π (z), ? z∈? p . First the distribution of linear transformations of Z are studied, and some moments of 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 Z and then investigate conditions for their independence. Finally explicit forms for the above distributions, m.g.f.’s and moments are derived when π (z)=κ (α′z), where α∈? p and κ is the normal, Laplace, logistic or uniform distribution function. |
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Keywords: | elliptical distribution independence moment-generating function multivariate skew-normal distribution quadratic form |
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