# Is there an analytic distribution for the sum of random variables distributed IID inverse gamma?

How about their ratio? I have looked at the related distributions section on Wikipedia and tried playing with the pdf's by hand.

I could have been a little more specific about the case that is currently of interest to me: Both inv-gamma distributions have the same parameters. Thinking about it, this is identical to asking about the sum of two realizations from the same inv-gamma distribution.

## 1 Answer

I doubt there's going to be a simple closed form for the distribution of the sum of inverse gamma variables in general (there might be some special cases that work out); you may be able to show it's equal to some series.

The ratio is something we can say more about.

If $X$ and $Y$ are inverse gamma, and $X_1=1/X$, $Y_1=1/Y$, then $X/Y = Y_1/X_1$, where the two subscripted variates are gamma distributed.

When their scale parameters are identical as here, that ratio is distributed as beta-prime (sometimes called a 'beta distribution of the second kind'). A simple rescaling of that produces an F distribution, so it's also a scaled F, but since the shape parameters are identical here, the scale factor should be 1.

Which is to say, if you have both shape parameters being $\alpha$ in the original inverse gamma distributions, you should get an $F(2\alpha,2\alpha)$ for the ratio.

[If the scale parameters are not identical, you can take the ratio of their scale parameters out as a multiplicative constant, and leave two gammas with identical scales -- so the ratio then has a scaled beta-prime distribution; again also a scaled F, though in this case the scale factor will be different from the one for the beta prime.]

Now that I've looked, I'm astonished that the relation to both the beta prime and the F is not listed on at least the Gamma distribution page. Both pieces of information are on the beta prime page.

• Glen, thanks for your answer. I edited the question because the 4th paragraph of your answer reminded me of a way I could make the question more specific. Does that edit to my question change your thoughts? – rcorty Mar 30 '15 at 17:31
• I took the "same parameters" as implied by IID (in the title), so the "as here" in my para. 4 refers to your question as it stands. I then briefly generalize (since you're not the only person this answer might help) to the case where the scale parameters may differ, and then return to that generalization again in my second-last paragraph. So no, nothing changes, that was the main case discussed in my answer. In the identical parameter case, the ratio should be F (or beta-prime) as mentioned, and I don't think the sum has any nice closed form. – Glen_b Mar 30 '15 at 20:50