Quadratic forms of multivariate skew normal-symmetric distributions
Following the paper by Gupta and Chang (Multivariate skew-symmetric distributions. Appl. Math. Lett. 16, 643-646 2003.) we generate a multivariate skew normal-symmetric distribution with probability density function of the form fZ(z)=2[phi]p(z;[Omega])G([alpha]'z), where , [phi]p(z;[Omega]) is the p-dimensional normal p.d.f. with zero mean vector and correlation matrix [Omega], and G is taken to be an absolutely continuous function such that G' is symmetric about 0. First we obtain the moment generating function of certain quadratic forms. It is interesting to find that the distributions of some quadratic forms are independent of G. Then the joint moment generating functions of a linear compound and a quadratic form, and two quadratic forms, and conditions for their independence are given. Finally we take G to be one of normal, Laplace, logistic or uniform distribution, and determine the distribution of a special quadratic form for each case.
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Volume (Year): 76 (2006)
Issue (Month): 9 (May)
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- A. Azzalini & A. Capitanio, 1999. "Statistical applications of the multivariate skew normal distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 579-602.
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- Genton, Marc G. & He, Li & Liu, Xiangwei, 2001. "Moments of skew-normal random vectors and their quadratic forms," Statistics & Probability Letters, Elsevier, vol. 51(4), pages 319-325, February.
- Arjun Gupta & Truc Nguyen & Jose Sanqui, 2004. "Characterization of the skew-normal distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(2), pages 351-360, June.
- Nadarajah, Saralees & Kotz, Samuel, 2003. "Skewed distributions generated by the normal kernel," Statistics & Probability Letters, Elsevier, vol. 65(3), pages 269-277, November.
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