A Distribution on the Simplex Arising from Inverted Chi-square Random Variables with Odd Degrees of Freedom

Jose Maria Del Castillo

Communications in Statistics - Theory and Methods, 2021, vol. 50, issue 4, 890-909

Abstract: We consider the random variables Zi=βi2Yi/∑k=1mβk2Yk where Yi,i=1…,m are independent inverted chi-square r.v. with νi degrees of freedom. The probability density function of Z=(Z1,Z2,…Zm) is obtained. When νi, i=1…,m are odd, it is shown how to obtain in a fairly easy way a closed form expression for the expectation of log (∑k=1mβk2Yk). Differentiating this expression with respect to the βi, one can find the moments of the random variables Zi. For the particular case of odd degrees of freedom, closed form expressions for the pdf of the univariate and multivariate marginal distributions of Z are also derived. The distribution of Z may be an alternative to the Dirichlet distribution for modeling compositional data.

Date: 2021
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DOI: 10.1080/03610926.2019.1643481

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