Beta prime distribution

Beta prime distribution is defined for ${\displaystyle x>0}$  with two parameters α and β, having the probability density function:

${\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}}$ 

where B is the Beta function.

The cumulative distribution function is

${\displaystyle F(x;\alpha ,\beta )=I_{\frac {x}{1+x}}\left(\alpha ,\beta \right),}$ 

where I is the regularized incomplete beta function.

While the related beta distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in odds. The distribution is a Pearson type VI distribution.^[1]

The mode of a variate X distributed as ${\displaystyle \beta '(\alpha ,\beta )}$  is ${\displaystyle {\hat {X}}={\frac {\alpha -1}{\beta +1}}}$ . Its mean is ${\displaystyle {\frac {\alpha }{\beta -1}}}$  if ${\displaystyle \beta >1}$  (if ${\displaystyle \beta \leq 1}$  the mean is infinite, in other words it has no well defined mean) and its variance is ${\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}}$  if ${\displaystyle \beta >2}$ .

For ${\displaystyle -\alpha <k<\beta }$ , the k-th moment ${\displaystyle E[X^{k}]}$  is given by

${\displaystyle E[X^{k}]={\frac {\mathrm {B} (\alpha +k,\beta -k)}{\mathrm {B} (\alpha ,\beta )}}.}$ 

For ${\displaystyle k\in \mathbb {N} }$  with ${\displaystyle k<\beta ,}$  this simplifies to

${\displaystyle E[X^{k}]=\prod _{i=1}^{k}{\frac {\alpha +i-1}{\beta -i}}.}$ 

The cdf can also be written as

${\displaystyle {\frac {x^{\alpha }\cdot {}_{2}F_{1}(\alpha ,\alpha +\beta ,\alpha +1,-x)}{\alpha \cdot \mathrm {B} (\alpha ,\beta )}}}$ 

where ${\displaystyle {}_{2}F_{1}}$  is the Gauss's hypergeometric function ₂F₁ .

The beta prime distribution may also be reparameterized in terms of its mean μ > 0 and precision ν > 0 parameters (^[2] p. 36).

Consider the parameterization μ = α/(β-1) and ν = β- 2, i.e., α = μ( 1 + ν) and β = 2 + ν. Under this parameterization E[Y] = μ and Var[Y] = μ(1 + μ)/ν.

Two more parameters can be added to form the generalized beta prime distribution ${\displaystyle \beta '(\alpha ,\beta ,p,q)}$ :

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

having the probability density function:

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

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.

This generalization can be obtained via the following invertible transformation. If ${\displaystyle y\sim \beta '(\alpha ,\beta )}$  and ${\displaystyle x=qy^{1/p}}$  for ${\displaystyle q,p>0}$ , then ${\displaystyle x\sim \beta '(\alpha ,\beta ,p,q)}$ .

The compound gamma distribution^[3] is the generalization of the beta prime when the scale parameter, q is added, but where p = 1. It is so named because it is formed by compounding two gamma distributions:

${\displaystyle \beta '(x;\alpha ,\beta ,1,q)=\int _{0}^{\infty }G(x;\alpha ,r)G(r;\beta ,q)\;dr}$ 

where ${\displaystyle G(x;a,b)}$  is the gamma pdf with shape ${\displaystyle a}$  and inverse scale ${\displaystyle b}$ .

The mode, mean and variance of the compound gamma can be obtained by multiplying the mode and mean in the above infobox by q and the variance by q².

Another way to express the compounding is if ${\displaystyle r\sim G(\beta ,q)}$  and ${\displaystyle x\mid r\sim G(\alpha ,r)}$ , then ${\displaystyle x\sim \beta '(\alpha ,\beta ,1,q)}$ . This gives one way to generate random variates with compound gamma, or beta prime distributions. Another is via the ratio of independent gamma variates, as shown below.

• If ${\displaystyle X\sim \beta '(\alpha ,\beta )}$  then ${\displaystyle {\tfrac {1}{X}}\sim \beta '(\beta ,\alpha )}$ .
• If ${\displaystyle Y\sim \beta '(\alpha ,\beta )}$ , and ${\displaystyle X=qY^{1/p}}$ , then ${\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}$ .
• 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 ${\displaystyle X\sim {\textrm {Beta}}(\alpha ,\beta )}$ , then ${\displaystyle {\frac {X}{1-X}}\sim \beta '(\alpha ,\beta )}$ . This property can be used to generate beta prime distributed variates.
• If ${\displaystyle X\sim \beta '(\alpha ,\beta )}$ , then ${\displaystyle {\frac {X}{1+X}}\sim {\textrm {Beta}}(\alpha ,\beta )}$ . This is a corollary from the property above.
• 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, ${\displaystyle X\sim \beta '(\alpha ,\beta ,1,{\tfrac {\beta }{\alpha }})}$ .
• For gamma distribution parametrization I:
  • If ${\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})}$  are independent, then ${\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\theta _{1}}{\theta _{2}}})}$ . Note ${\displaystyle \alpha _{1},\alpha _{2},{\tfrac {\theta _{1}}{\theta _{2}}}}$  are all scale parameters for their respective distributions.
• For gamma distribution parametrization II:
  • If ${\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})}$  are independent, then ${\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\beta _{2}}{\beta _{1}}})}$ . The ${\displaystyle \beta _{k}}$  are rate parameters, while ${\displaystyle {\tfrac {\beta _{2}}{\beta _{1}}}}$  is a scale parameter.
  • If ${\displaystyle \beta _{2}\sim \Gamma (\alpha _{1},\beta _{1})}$  and ${\displaystyle X_{2}\mid \beta _{2}\sim \Gamma (\alpha _{2},\beta _{2})}$ , then ${\displaystyle X_{2}\sim \beta '(\alpha _{2},\alpha _{1},1,\beta _{1})}$ . The ${\displaystyle \beta _{k}}$  are rate parameters for the gamma distributions, but ${\displaystyle \beta _{1}}$  is the scale parameter for the beta prime.
• ${\displaystyle \beta '(p,1,a,b)={\textrm {Dagum}}(p,a,b)}$  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 ${\displaystyle {\dfrac {X}{x_{m}}}-1\sim \beta ^{\prime }(1,\alpha )}$ .
• 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.
• If ${\displaystyle X\sim \beta '(\alpha ,\beta )}$ , then ${\displaystyle \ln X}$  has a generalized logistic distribution. More generally, if ${\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)}$ , then ${\displaystyle \ln X}$  has a scaled and shifted generalized logistic distribution.

• Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0
• Bourguignon, M.; Santos-Neto, M.; de Castro, M. (2021), "A new regression model for positive random variables with skewed and long tail", Metron, 79: 33–55, doi:10.1007/s40300-021-00203-y, S2CID 233534544

• MathWorld article