Talk:Beta prime distribution

Beta Prime and F

The comment about Beta Prime and F appears to be incorrect. Although it makes sense that these are related the statement that if b is beta prime, then b*alpha/beta is F can't make sense, since if alpha = beta then you just get b and F distribution is not invariant when you multiply d1 and d2 by constants. —Preceding unsigned comment added by 83.244.153.18 (talk) 11:31, 7 April 2010 (UTC)

Cumulative distribution function and excess kurtosis

Dear main authors: as you probably know, the beta-prime distribution is the same as the F-distribution, if one replaces in the latter: $d_{1}/2\rightarrow \alpha$, $d_{2}/2\rightarrow \beta$, $d_{1}x/d_{2}\rightarrow x$. That means that part of the text in the F-distribution-article can be copied and pasted into this article. That's what I did for the CDF and the kurtosis. Regards: Herbmuell (talk) 17:59, 21 July 2015 (UTC).

Also on the relation between Beta Prime and F

${\tfrac {\alpha }{\beta }}$ should be flipped to get

If $X\sim F(2\alpha ,2\beta )$ has an F-distribution, then ${\tfrac {\alpha }{\beta }}X\sim \beta '(\alpha ,\beta )$, or equivalently, $X\sim \beta '(\alpha ,\beta ,1,{\tfrac {\beta }{\alpha }})$.

Alternative parameterization does not match parameters from sidebox

The sidebox of the article states that the mean and variance of the distributions are

$\mu ={\frac {\alpha }{\beta -1}}{\text{ if }}\beta >1$

and

$v={\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}{\text{ if }}\beta >2$

However, solving the system of equations for $\alpha$ and $\beta$ shows the alternate parameterization should be

$\alpha =\mu (\beta -1)$

and

$\beta ={\frac {\mu (\mu +1)}{v}}+2$

So the given $\beta$ solution in the text is incorrect — Preceding unsigned comment added by 198.208.46.92 (talk) 23:57, 20 June 2022 (UTC)

The sum of i.i.d. beta prime variables

Before posting this discussion, the following unsourced claim was made in the article:

• If $X_{1}\sim \beta '(\alpha ,\beta )$ and $X_{2}\sim \beta '(\alpha ,\beta )$ two iid variables, then $Y=X_{1}+X_{2}\sim \beta '(\gamma ,\delta )$ with $\gamma ={\frac {2\alpha (\alpha +\beta ^{2}-2\beta +2\alpha \beta -4\alpha +1)}{(\beta -1)(\alpha +\beta -1)}}$ and $\delta ={\frac {2\alpha +\beta ^{2}-\beta +2\alpha \beta -4\alpha }{\alpha +\beta -1}}$, as the beta prime distribution is infinitely divisible.
• More generally, let $X_{1},...,X_{n}n$ iid variables following the same beta prime distribution, i.e. $\forall i,1\leq i\leq n,X_{i}\sim \beta '(\alpha ,\beta )$, then the sum $S=X_{1}+...+X_{n}\sim \beta '(\gamma ,\delta )$ with $\gamma ={\frac {n\alpha (\alpha +\beta ^{2}-2\beta +n\alpha \beta -2n\alpha +1)}{(\beta -1)(\alpha +\beta -1)}}$ and $\delta ={\frac {2\alpha +\beta ^{2}-\beta +n\alpha \beta -2n\alpha }{\alpha +\beta -1}}$.

The origin of the claim appears to be from this Stack Exchange post. The assumption in the question and the result of the SE post are both incorrect. The faulty assumption is that infinite divisibility of the beta prime distribution implies the beta prime distribution is stable.

The general case doesn't seem to be easily computable, but specific counterexamples are reasonable to perform. Here I will evaluate the density function $f_{Z}(1)$ where $Z=X+X$ and $X\sim \beta '(1,3)$.

Let $f(x)$ be the density function of $X$, so that $f(x)=3(1+x)^{-4}$. The density function of $Z$ can be computed using the convolution of probability distributions. I will omit the finer details, but you can determine $$f_{Z}(z)=\int _{0}^{z}f(x)\cdot f(z-x)\,dx=9\int _{0}^{z}(1+x)^{-4}(1+z-x)^{-4}\,dx.$$ This is computable for all z, but the specific case $z=1$ gives $f_{Z}(1)={\tfrac {291+160\log(2)}{972}}$.

Were the original claim to be true, then we would have $Z\sim \beta '({\tfrac {7}{3}},{\tfrac {10}{3}})$ which gives $f_{Z}(1)={\tfrac {1}{2^{17/3}\cdot \mathrm {B} (7/3,10/3)}}$. These two numbers are not equal.

I will remove the claims from the article. HailSaturn (talk) 03:03, 18 November 2024 (UTC)