Beta prime distribution

Two more parameters can be added to form the generalized beta prime distribution.

• ${\displaystyle p>0}$ shape (real)
• ${\displaystyle q>0}$ scale (real)

having the probability density function:

${\displaystyle f(x;\alpha ,\beta ,p,q)={\frac {p\left({\frac {x}{q}}\right)^{\alpha p-1}\left(1+\left({\frac {x}{q}}\right)^{p}\right)^{-\alpha -\beta }}{qB(\alpha ,\beta )}}}$

with mean

${\displaystyle {\frac {q\Gamma \left(\alpha +{\tfrac {1}{p}}\right)\Gamma (\beta -{\tfrac {1}{p}})}{\Gamma (\alpha )\Gamma (\beta )}}\quad {\text{if }}\beta p>1}$

and mode

${\displaystyle q\left({\frac {\alpha p-1}{\beta p+1}}\right)^{\tfrac {1}{p}}\quad {\text{if }}\alpha p\geq 1}$

Note that if p = q = 1 then the generalized beta prime distribution reduces to the standard beta prime distribution

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

• If ${\displaystyle X\sim F(2\alpha ,2\beta )}$ has an F-distribution, then ${\displaystyle {\tfrac {\alpha }{\beta }}X\sim \beta '(\alpha ,\beta )}$, or equivalently, ${\displaystyle X\sim \beta '(\alpha ,\beta ,1,{\tfrac {\beta }{\alpha }})}$.
• If ${\displaystyle X\sim {\textrm {Beta}}(\alpha ,\beta )}$ then ${\displaystyle {\frac {X}{1-X}}\sim \beta '(\alpha ,\beta )}$.
• If ${\displaystyle X\sim \Gamma (\alpha ,1)}$ and ${\displaystyle Y\sim \Gamma (\beta ,1)}$ are independent, then ${\displaystyle {\frac {X}{Y}}\sim \beta '(\alpha ,\beta )}$.
• Parametrization 1: If ${\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})}$ are independent, then ${\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\theta _{1}}{\theta _{2}}})}$.
• Parametrization 2: If ${\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})}$ are independent, then ${\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\beta _{2}}{\beta _{1}}})}$.
• ${\displaystyle \beta '(p,1,a,b)={\textrm {Dagum}}(p,a,b)}$ the Dagum distribution
• ${\displaystyle \beta '(1,p,a,b)={\textrm {SinghMaddala}}(p,a,b)}$ the Singh–Maddala distribution.
• ${\displaystyle \beta '(1,1,\gamma ,\sigma )={\textrm {LL}}(\gamma ,\sigma )}$ the log logistic distribution.
• The beta prime distribution is a special case of the type 6 Pearson distribution.
• If X has a Pareto distribution with minimum ${\displaystyle x_{m}}$ and shape parameter ${\displaystyle \alpha }$, then ${\displaystyle X-x_{m}\sim \beta ^{\prime }(1,\alpha )}$.
• If X has a Lomax distribution, also known as a Pareto Type II distribution, with shape parameter ${\displaystyle \alpha }$ and scale parameter ${\displaystyle \lambda }$, then ${\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )}$.
• If X has a standard Pareto Type IV distribution with shape parameter ${\displaystyle \alpha }$ and inequality parameter ${\displaystyle \gamma }$, then ${\displaystyle X^{\frac {1}{\gamma }}\sim \beta ^{\prime }(1,\alpha )}$, or equivalently, ${\displaystyle X\sim \beta ^{\prime }(1,\alpha ,{\tfrac {1}{\gamma }},1)}$.
• The inverted Dirichlet distribution is a generalization of the beta prime distribution.

• Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0
• MathWorld article