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Quadratic forms of multivariate skew normal-symmetric distributions

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  • Huang, Wen-Jang
  • Chen, Yan-Hau
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    Abstract

    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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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 76 (2006)
    Issue (Month): 9 (May)
    Pages: 871-879

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    Handle: RePEc:eee:stapro:v:76:y:2006:i:9:p:871-879

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    Related research

    Keywords: Chi-square distribution Independence Moment generating function Non-normal models Quadratic form Skew normal distribution Skew symmetric distribution;

    References

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    Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
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    1. Arjun Gupta & Truc Nguyen & Jose Sanqui, 2004. "Characterization of the skew-normal distribution," Annals of the Institute of Statistical Mathematics, Springer, vol. 56(2), pages 351-360, June.
    2. Loperfido, Nicola, 2001. "Quadratic forms of skew-normal random vectors," Statistics & Probability Letters, Elsevier, vol. 54(4), pages 381-387, October.
    3. Gupta, Arjun K. & González-Farías, Graciela & Domínguez-Molina, J. Armando, 2004. "A multivariate skew normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 89(1), pages 181-190, April.
    4. Arnold, Barry C. & Beaver, Robert J., 2000. "The skew-Cauchy distribution," Statistics & Probability Letters, Elsevier, vol. 49(3), pages 285-290, September.
    5. 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.
    6. 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.
    7. 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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    Cited by:
    1. Huang, Wen-Jang & Chen, Yan-Hau, 2007. "Generalized skew-Cauchy distribution," Statistics & Probability Letters, Elsevier, vol. 77(11), pages 1137-1147, June.

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