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Remarks for the singular multivariate skew-normal distribution and its quadratic forms

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  • Li, Baokun
  • Tian, Weizhong
  • Wang, Tonghui

Abstract

Under the singular multivariate skew normal (SMSN) setting, we showed that, in this paper, the necessary and sufficient conditions for independence of two sub-vectors given in Young et al. (2017) are equivalent to the results in Wang et al. (2009). In addition, the distribution of quadratic form of SMSN random vector is derived, with this new definition of the noncentral skew chi-square distribution. Several examples are given to illustrate our main results.

Suggested Citation

  • Li, Baokun & Tian, Weizhong & Wang, Tonghui, 2018. "Remarks for the singular multivariate skew-normal distribution and its quadratic forms," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 105-112.
    • Handle: RePEc:eee:stapro:v:137:y:2018:i:c:p:105-112
    DOI: 10.1016/j.spl.2018.01.008
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    References listed on IDEAS

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    1. Young, Phil D. & Harvill, Jane L. & Young, Dean M., 2016. "A derivation of the multivariate singular skew-normal density function," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 40-45.
    2. Wang, Tonghui & Li, Baokun & Gupta, Arjun K., 2009. "Distribution of quadratic forms under skew normal settings," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 533-545, March.
    3. 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.
    4. Ye, Rendao & Wang, Tonghui & Gupta, Arjun K., 2014. "Distribution of matrix quadratic forms under skew-normal settings," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 229-239.
    5. Young, Phil D. & Kahle, David J. & Young, Dean M., 2017. "On the independence of singular multivariate skew-normal sub-vectors," Statistics & Probability Letters, Elsevier, vol. 122(C), pages 58-62.
    6. Vernic, Raluca, 2006. "Multivariate skew-normal distributions with applications in insurance," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 413-426, April.
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