Distribution of the ratio of Dirichlet/Gamma variates

Question

It can be seen that the following random variates have the same distribution:

1. $\frac{X_1 + X_3}{X_2 + X_3}$, where $(X_1, X_2, X_3) \sim \text{Dirichlet} (\alpha_1, \alpha_2, \alpha_3)$
2. $\frac{Y_1 + Y_3}{Y_2 + Y_3}$, where $(Y_0, Y_1, Y_2, Y_3) \sim \text{Dirichlet} (\alpha_0, \alpha_1, \alpha_2, \alpha_3)$
3. $\frac{Z_1 + Z_3}{Z_2 + Z_3}$, where the $(Z_i)_i$ are independent and $Z_i \sim \text{Gamma}(k = \alpha_i, \theta = 1)$

Question: does this distribution have a name? Has it been studied somewhere in the literature? Were it not for $X_1$ in the numerator, it seems that this would be a Beta-Prime distribution.

Answer 1

It turns out that yes, this distribution has indeed been studied:

Lee, Ru-Ying, Burt S. Holland, and John A. Flueck. "Distribution of a ratio of correlated gamma random variables." SIAM Journal on Applied Mathematics 36.2 (1979): 304-320.

The authors have not given a name to this 3-parameters family of distribution. They frame this in the context of the Cherian-David-Fix bivarate structure.

I would personally suggest to call this a Dirichlet Process Mass Ratio distribution, denoted $\text{DPMR}(\alpha_1, \alpha_2, \alpha_3)$, reflecting the fact that such a random variable is the ratio of (random) probability masses given to two sets $S_1$ and $S_2$ by a Dirichlet Process. $\alpha_1$ corresponds to $S_1-S_2$, $\alpha_2$ to $S_2-S_1$, and $\alpha_3$ to $S_1 \cap S_2$, consistently with writing these indices in binary (1 -> 01, 2 -> 10, 3 -> 11).