# Inverted Dirichlet distribution

In statistics, the inverted Dirichlet distribution is a multivariate generalization of the beta prime distribution, and is related to the Dirichlet distribution. It was first described by Tiao and Cuttman in 1965.[1]

The distribution has a density function given by

$p\left(x_1,\ldots, x_k\right) = \frac{\Gamma\left(\nu_1+\cdots+\nu_{k+1}\right)}{\prod_{j=1}^{k+1}\Gamma\left(\nu_j\right)} x_1^{\nu_1-1}\cdots x_k^{\nu_k-1}\times\left(1+\sum_{i=1}^k x_i\right)^{-\sum_{j=1}^{k+1}\nu_j},\qquad x_i>0.$

The distribution has applications in statistical regression and arises naturally when considering the multivariate Student distribution. It can be characterized[2] by its moment generating function:

$E\left[\prod_{i=1}^kx_i^{q_i}\right] = \frac{\Gamma\left(\nu_{k+1}-\sum_{j=1}^k\nu_j\right)}{\Gamma\left(\nu_{n+1}\right)}\prod_{j=1}^k\frac{\Gamma\left(\nu_j+q_j\right)}{\Gamma\left(\nu_j\right)}$

provided that $q_j>-\nu_j, 1\leqslant j\leqslant k$ and $\nu_{n+1}>q_1+\ldots+q_k$.

The inverted Dirichlet distribution is conjugate to the negative multinomial distribution if a generalized form of odds ratio is used instead of the categories' probabilities.

## References

1. ^ Tiao, George T. (1965). "The inverted Dirichlet distribution with applications". Journal of the American Statistical Association 60 (311): 793–805.
2. ^ Ghorbel, M. (2010). "On the inverted Dirichlet distribution". Communications in Statistics---Theory and Methods 39: 21–37.