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Quadratic forms of refined skew normal models based on stochastic representation. (English) Zbl 1375.60051
Summary: In [J. Multivariate Anal. 100, No. 3, 533–545 (2009; Zbl 1154.62342)], the second author et al. first introduced the skew chi-square distribution based on the multivariate skew normal distribution provided by A. Azzalini [Scand. J. Stat. 12, 171–178 (1985; Zbl 0581.62014)], and R. Ye et al. [J. Multivariate Anal. 131, 229–239 (2014; Zbl 1298.62086)] extended this results into the skew Wishart distribution. Motivated by these results, we first study a new type of multivariate skew normal distribution introduced by A. K. Gupta and J. T. Chen [Ann. Inst. Stat. Math. 56, No. 2, 305–315 (2004; Zbl 1056.62064)], the moment generating function, independence and quadratic form are discussed, and also a new type of skew chi-square distribution was introduced. Later on, we defined a new type of skew Wishart distribution based on the matrix skew normal models introduced by W. Ning [Random Oper. Stoch. Equ. 23, No. 1, 21–29 (2015; Zbl 1310.62065)]. In the end, we will study the probabilistic representation of multivariate skew elliptical models.

MSC:
60E05 Probability distributions: general theory
62H05 Characterization and structure theory for multivariate probability distributions; copulas
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