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(it's the first time that I write here, sorry if miss some convention)

If X ~ Gamma(a, b) then 1/X ~ inv_Gamma(a, b)

therefore if

If X ~ Gamma(a, b) then 1/X^r ~ inv_Gamma(?, ?)

My brain is exploding I don't have a great math skills unfortunately.

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    $\begingroup$ The greetings didn't get through, that's a good convention to have for sure ;) $\endgroup$ – Stefano Vespucci May 4 '17 at 14:33
  • $\begingroup$ If $X^{-r}$ were inverse gamma, it would imply $X^{r}$ is gamma. But that's not the case. $\endgroup$ – Glen_b May 5 '17 at 6:38
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If $X \sim G \left( a,b \right)$ is a gamma random variable with shape parameter $a$ and scale parameter $b$ with mean $ab,$ then $X^r$ (where $r>0$) is a generalized gamma random variable with three parameters. The first shape parameter is $a,$ the scale parameter will be $b^r,$ and the second shape parameter is $1 \over r$ .

We write it as $Y=X^r \sim GG \left(a,b^r,{1 \over r} \right)$

There is more detail given here: http://mathscience.kiau.ac.ir/Content/Vol4No1/2.pdf

The reciprocal of $Y$ has a generalized inverse gamma distribution. A reference for that is here: https://arxiv.org/pdf/1005.3274.pdf

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  • $\begingroup$ The last reference is what I was looking for, now I have to implement this is Stan. I wish they had such distribution $\endgroup$ – Stefano Vespucci May 5 '17 at 0:02
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Did you check Wikipedia? Answers can be found there, but must be put together from various pages. First, https://en.wikipedia.org/wiki/Inverse-gamma_distribution the inverse gamma distribution is the distribution of the inverse $1/X$ when $X$ has a gamma distribution, so by reading that definition in reverse, the inverse of an inverse gamma random variable must have a gamma distribution. You ask for the distribution of the $r$th power of that, that is, the $r$th power of a gamma random variable. According to https://en.wikipedia.org/wiki/Gamma_distribution#Related_distributions_and_properties that have a generalized gamma distribution, which you can read about here: https://en.wikipedia.org/wiki/Generalized_gamma_distribution

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