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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?

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    $\begingroup$ Isn't this just the generalized gamma? (Generalized gammas are gammas to a power; the chi-square is gamma to a power.) $\endgroup$
    – shabbychef
    May 29, 2015 at 22:12

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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

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    $\begingroup$ 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". $\endgroup$
    – Christoph Hanck
    Mar 25, 2015 at 15:54
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    $\begingroup$ 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... $\endgroup$
    – Ben
    Mar 25, 2015 at 16:24
  • $\begingroup$ 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). $\endgroup$
    – ben18785
    Mar 25, 2015 at 16:30
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    $\begingroup$ I would really like to give you an up vote but I can't since I need 15 reputation :( !? $\endgroup$
    – Ben
    Mar 25, 2015 at 16:32
  • $\begingroup$ 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. $\endgroup$
    – Christoph Hanck
    Mar 25, 2015 at 16:53

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