# Square root of an inverse gamma distributed random variable

I work on the grouped t copula and try to replicate part of the following paper: "The t copula with Multiple Parameters of Degrees of Freedom: Bivariate Characteristics and Application to Risk Management", Luo & Shevchenko (2010).

My Question:

I need the distribution of the following random variable: $$Y = \sqrt{\frac\nu X}$$ where X is Chi square distributed with v degrees of freedom.

I found out that the following random variabe Z is inverse gamma distributed with alpha = v/2 and beta = v/2, $$Z = {\frac\nu X}$$

So, basically I am looking for the distribution of the square root of an inverse gamma distributed random variable.

Do you have any ideas?

• Isn't this just the generalized gamma? (Generalized gammas are gammas to a power; the chi-square is gamma to a power.) May 29 '15 at 22:12

I have derived the answer using Mathematica:

PDF[TransformedDistribution[Sqrt[x], x \[Distributed] InverseGammaDistribution[\[Nu]/2, \[Nu]/2]], x]


results in a pdf for your transformed variable of the form:

$\frac{2^{1-\frac{\nu}{2}} e^{-\frac{\nu}{2x^2}}x^{-1-\nu}\nu^{\nu/2}}{Gamma(\nu/2)}$

I am not sure if this represents a particular named distribution, but hope that knowing the pdf may help in some way.

Update: the inverse cdf for this distribution is:

$\frac{\nu}{2 InverseGammaRegularised(\nu/2,x)}$

Best,

Ben

• If $Z$ is $IG$, we are looking for the distribution of $$\sqrt{\frac{1}{X}}=\frac{1}{\sqrt{X}},$$ where $X$ follows a Gamma-distribution, right? Apparently, the distribution of $\sqrt{X}$ is known as the Nagakami distribution. So, a possible name would be "inverse Nagakami distribution". Mar 25 '15 at 15:54
• What I need is the quantile function / inverse cdf of this distribution, since I want to use it in R. @ Ben: Is it possible to derive this with Mathematica? @ Chrisoph: Doesn't $$\sqrt X$$ in your example follow a generalized gamma distribution? The Nagakami distribution is already implemented in R but I am not sure how I could use this for my problem...
– Ben
Mar 25 '15 at 16:24
• Ok, I have updated my answer to include this result (I'm not asking for an up vote, but as Christoph says, it is typically the way to thank people). Mar 25 '15 at 16:30
• I would really like to give you an up vote but I can't since I need 15 reputation :( !?
– Ben
Mar 25 '15 at 16:32
• I am not sure about the generalized one. Why do you think so? I am also not sure if you can use my result. You could try to apply the change of variables rule for r.v.s to check @ben18785 's finding by trying to derive the distribution of $1/W$ if $W$ follows a Nagakami distribution. Mar 25 '15 at 16:53