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

Abstract

We consider the random variables $Z_i={\beta }_i^2{Y}_i/{\sum }_{k=1}^m{\beta }_k^2{Y}_k$ where $Y_i,i=1\dots ,m$ are independent inverted chi-square r.v. with ν_i degrees of freedom. The probability density function of $\mathbf{Z}=(Z_1,Z_2,\dots Z_m)$ is obtained. When ${\nu }_i, i=1\dots ,m$ are odd, it is shown how to obtain in a fairly easy way a closed form expression for the expectation of $\text{log}\hspace{0.17em}({\sum }_{k=1}^m{\beta }_k^2{Y}_k).$ Differentiating this expression with respect to the β_i, one can find the moments of the random variables Z_i. 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.

Keywords:

• Inverted chi-square distribution
• chi-square distribution
• Dirichlet distribution
• Bessel polynomials