Talk:Inverse-gamma distribution

K_alpha function

In characteristic function for this distribution, there is a function K_alpha. What is this? What is it called? —Preceding unsigned comment added by 95.72.39.82 (talk) 22:34, 2 March 2009 (UTC)

mfg change

I changed the mfg to "Does not exist" because it doesn't exist. Richard Finlay

Basmandude 16:05, 27 July 2006 (UTC) What the heck is going on with thi page? Gone all funky...

User:Helenuh 10:05 27 Nov 2006 The relation between X and Y in the relation between inverse-gamma and inverse-chisquare is unclear. I think it should be Y=X

CDF figure incorrect?

The gnuplot source for the CDF figure is given as:

pinvgamma(x,a,b) = 1 - igamma(a, b*1.0/x)

but it seems to me that the CDF should actually be:

pinvgamma(x,a,b) = 1 - igamma(a, b*1.0/x) / gamma(a)

if igamma is the lower incomplete gamma function, since we have:

${\frac {\Gamma (\alpha ,x)}{\Gamma (a)}}=1-{\frac {\gamma (\alpha ,x)}{\Gamma (a)}}$

from Incomplete gamma function. Ged.R 12:03, 8 March 2007 (UTC)

Scaled inverse gamma distribution

Shouldn't there be a reference to the scaled inverse gamma distribution? I am not a statistitian so I do not feel comfortable explaining the differences. The difference is just a transformation or a substitution? Thanks. --Kupirijo 16:30, 31 March 2007 (UTC)

naming conventions

I think it should be $\tau$ instead of $\sigma$ since $\sigma$ is usually used for standard deviations but here it is used as precision, which is usually labeled $\tau$. 129.26.160.2 11:39, 14 September 2007 (UTC)

Related Distributions Typo?

Maybe I'm missing something obvious but shouldn't it be

If $X\sim {\mbox{Inv-Gamma}}(k,\theta )\,,$ then $1/X\sim {\mbox{Gamma}}(k,\theta )\,,$ is a Gamma distribution.

Thenegus 13:15, 28 September 2007 (UTC)Thenegus

You are right it should be the reciprocal of the theta in original Gamma distribution. The French page gets this correct.

"If $X\sim {\mbox{Inv-Gamma}}(k,\theta )\,,$ then $1/X\sim {\mbox{Gamma}}(k,1/\theta )\,,$ is a Gamma distribution."

-Richard_Ren 14:50 16 May 2012

I think the $X\sim {\mbox{Inv-Gamma}}(k,\theta )\,,$ then $1/X\sim {\mbox{Gamma}}(k,1/\theta )\,,$ version is correct. See Section 2 here

http://www.johndcook.com/inverse_gamma.pdf

for a derivation.

--Dan Greenwald

Thenegus is right. There is mistake in the derivation in the document provided by Dan Greenwald. In first term, gamma function is inverse $1/\Gamma (\alpha )$(as in gamma distribution), while beta to the power of alpha somehow becomes inverse $1/\beta ^{\alpha }$ instead of $\beta ^{\alpha }$.

The derivation in the current version of wiki article is correct, but it is confused by notation. It starts with $(k,\theta )$, but in last line goes back to $(\alpha ,\beta )$. But since $1/\theta =\beta$, it means

If $X\sim {\mbox{Gamma}}(\alpha ,\beta )\,,$ then $1/X\sim {\mbox{Inv-Gamma}}(\alpha ,\beta )$

You can also get confirmation of this by comparing mean and variance of both gamma and inverse gamma distribution.

--- Eugene ---- 25 June 2012

The article needs some cleaning

The article is not really a good one and does not quite give the inverse-gamma the credit its due. The formula for the density was wrong; the correct version is proportional to (1/x)^{\alpha + 1}\exp(-\beta/x). The `moment-generating function' is a bit odd to include, as it does not exist for any positive t (only for negative t, and zero); it's better to erase it and to include the Laplace transform. Also, it is mildly dangerous to give a formula in terms of a certain `K_\alpha' function without any further pointer to what this function actually is. Slavatrudu (talk) 18:19, 24 April 2008 (UTC)

I agree that there are problems with the article. I'm not prepared to say that Slavatrudu is correct in pointing out that the density function is incorrect (a quick check of external sources suggests that the density equation currently given is correct), but it does disagree with the plots. 66.117.129.43 (talk) 18:56, 17 November 2008 (UTC)

mgf @ cf

Does K_\alpha mean the Bessel function? But what concerns the moment generating function, this is wrong, since it does not exist. —Preceding unsigned comment added by 194.145.96.51 (talk) 09:47, 4 June 2010 (UTC)

should the parameters alpha and beta range include zero

should the parameters alpha and beta include zero

so alpha >= 0 rather than alpha > 0

This seems consistent with the supported range?

And a velue of zero seems reasonable. —Preceding unsigned comment added by Pabristow (talk • contribs) 13:59, 3 September 2010 (UTC)

An inverse gamma disribution function for alpha=0 does not exist, because in order for it to be normalised, we need to evaluate the integral corresponding to a gamma function evaluated at 0. This is undefined, hence inverse gamma is not a proper probability distribution for alpha=0 (Probabilityislogic (talk) 13:12, 24 January 2012 (UTC))

What is meant by beta being the scale parameter with respect to the inverse gamma distirbution?

In the "Derivation from the gamma distribution" section, the page notes

>Note that beta is the scale parameter from the perspective of the inverse gamma distribution.

This seems to be unclear without additional justification. What is meant by this exactly? Phdemetri (talk) 20:43, 26 August 2022 (UTC)

@Phdemetri: A parameter θ of a family of distributions is a "scale parameter" if the following holds. If two variables x and y have distributions from that family, differing only in having different values of θ, say θ₁ and θ₂, then the distribution of x is the same as that of y "scaled" by being multiplied by the factor θ₁/θ₂. Probably the best known example is the standard deviation σ in the case of a normal distribution: changing the value of σ scales the distribution without changing its shape. Evidently "β is the scale parameter from the perspective of the inverse gamma distribution" is intended to mean that β serves as a scale factor in the inverse gamma distribution, whereas it is not a scale parameter in the gamma distribution. As with a lot of mathematical articles on Wikipedia, this was no doubt written by a mathematician writing for other mathematicians with a similar background knowledge of the same subject area, without regard for the general reader. It could no doubt be better written, but I don't have time to do that now. I may or may not come back to it. JBW (talk) 21:06, 26 August 2022 (UTC)

I made a small edit to the page to demonstrate that beta satisfies the conditions for being called a scale parameter, linking to the scale parameter wiki page. Feel free to edit it if you feel necessary for content or formatting. Phdemetri (talk) 22:24, 26 August 2022 (UTC)