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What is the distribution (probability density function) of the difference of two Gammas with the same scale? The parameters of each Gamma would be positive integers.

That is, I'd like to know the density function ($f$) of $Z=X-Y$, where $X\sim {\rm Gamma}(a_1,b_1)$ and $Y\sim {\rm Gamma}(a_2, b_1)$. Assume $c$, $d$ are the parameters of $f(Z)$, particularly, I want to know the relationships between $c$, $d$, and $a_1$, $a_2$, $b_1$.

gung - Reinstate Monica
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jxu
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  • Because Gamma distributions are Chi-squared distributions, my answer at http://stats.stackexchange.com/questions/51699 gives the solution, identifying it as a variance-gamma distribution. – whuber Jan 30 '17 at 00:19
  • @whuber, should we close this as a duplicate, then? – gung - Reinstate Monica Jan 30 '17 at 00:31
  • @Gung It would be a duplicate only to people who easily recognize the similarities, so I am inclined to be cautious about closing. – whuber Jan 30 '17 at 00:33
  • @jxu Are you seeking a name for this or a functional form for the density? – Glen_b Jan 30 '17 at 01:03
  • @whuber I read link but it is still not straightforward to me. I'd like to know the density function (f) of Z=X-Y, where X~Gamma(a1,b1) and Y~Gamma(a2,b1). Assume c, d are the parameters of f(Z), particularly, I want to know the relationships between c, d and a1,a2,b1. Thanks. – jxu Jan 30 '17 at 02:03
  • My post does not fully answer your question: it treats only the case of identical shape and scale parameters. The nature of the result (which includes a Bessel function) indicates this is unlikely to have a simple expression. It's unusual to consider differences of Gamma distributions (or, for that matter, differences of positive distributions) because differences are used to compare means--but the most natural method to compare means of gammas is by taking their ratio rather than their difference. That creates a [Beta prime distribution](https://en.wikipedia.org/wiki/Beta_prime_distribution). – whuber Jan 30 '17 at 14:59
  • Considering differences of Gamma distributions is more meaningful than ratio for my purpose. Thanks for pointing out that there is probably no simple closed form. I will turn to MC. Thanks. – jxu Jan 30 '17 at 19:41

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