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)=2φp(z;Ω)G(α′z), where Ω>0,α∈Rp, φp(z;Ω) is the p-dimensional normal p.d.f. with zero mean vector and correlation matrix Ω, 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.