# Mean of Generalization of the Dirichlet Distribution

I know that if $$X_{1},X_{2},...X_{n}$$ are independent $$\mathrm{Gamma}(\alpha_{i},\theta)$$ - distributed variables (notice they all have the same scale parameter $$\theta$$) and

$$Y_{i}=\frac{X_{i}}{\sum_{j=1}^{n}X_{j}}$$

then :

$$Y=(Y_{1},Y_{2},...Y_{n})\;$$~$$\;\mathrm{Dirichlet}(\alpha_1,\alpha_2,...,\alpha_n)$$

I'm interested in what happens if $$X_{1},X_{2},...X_{n}$$ are allowed to have different scale parameters. That is:

$$X_i$$~$$\mathrm{Gamma}(\alpha_i,\theta_i)$$

The problem is: find a closed-form solution for the expectation of $$Y_i\ \; \forall i$$.

Great if you can also tell me how this distribution is called and provide a reference (Textbook, paper).

Bounty's on. Thank you!

• Aug 3, 2021 at 21:43
• There probably is no closed form for the mean and no common name for the distribution. Aug 4, 2021 at 12:54
• There actually is a common name: the scaled Dirichlet. See researchgate.net/publication/… (which addresses your question about the lack of closed form solution for the expectations) and Chapter 10, "Notes on the Scaled Dirichlet Distribution" in Compositional Data Analysis: Theory and Applications, ed. Vera Pawlowsky-Glahn and Antonella Buccianti. Jul 19, 2022 at 17:06
• There are at least 3 questions about this type of distribution. It would be great to have a tag to identify them easily: scaled-dirichlet-distribution. Could somebody with the requisite points kindly create such a tag? TIA. Jul 20, 2022 at 18:19

As mentioned by Matt F. in the comments, there is highly unlikely to be a closed-form solution. To see why, consider that the denominator's density itself "barely" has a closed-form, with a density given by the infinite series set out in equation 2.9 in Moschopoulos (1985) https://doi.org/10.1007/BF02481056. Specifically, the sum of n gamma distributions is given by: \begin{aligned} g(y)=C\sum_{k=0}^\infty\frac{\delta_k y^{\rho+k-1}e^{-y/{\beta_1}}}{\Gamma(\rho+k)\beta_1^{\rho+k}}\\ \end{aligned}
Here, $$\{\delta_k\}$$ is an infinite series of constants, $$\rho$$ and $$C$$ are also constants, and $$\beta_1$$ is the smallest value of $$\theta_i$$
• Thanks! I thought maybe at first, but I'm no longer so sure. Let $Z=\sum X_i$. Since both the variance and expectation of $1/Z$ are declining with $n$, it's not at all clear to me that there is a nice limit. If I were going to try to show this, I might try to prove that for large $n$, $cov[X_i,1/Z]\to 0$ much faster than $E[X_i]E[1/Z]\to 0$, since then then $E[X_i/Z] \approx E[X_i]E[1/Z]$. But it is not at all clear to me that this is true. Aug 8, 2021 at 20:06