Beta prime distribution

In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind) is an absolutely continuous probability distribution. If $p\in [0,1]$ has a beta distribution, then the odds ${\frac {p}{1-p}}$ has a beta prime distribution.

Beta prime

Probability density function
Cumulative distribution function
Parameters	$\alpha >0$ shape (real) $\beta >0$ shape (real)
Support	$x\in [0,\infty )\!$
PDF	$f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}\!$
CDF	$I_{{\frac {x}{1+x}}(\alpha ,\beta )}$ where $I_{x}(\alpha ,\beta )$ is the incomplete beta function
Mean	${\frac {\alpha }{\beta -1}}$ if $\beta >1$
Mode	${\frac {\alpha -1}{\beta +1}}{\text{ if }}\alpha \geq 1{\text{, 0 otherwise}}\!$
Variance	${\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}$ if $\beta >2$
Skewness	${\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}$ if $\beta >3$
Excess kurtosis	$6{\frac {\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}}$ if $\beta >4$
Entropy	${\begin{aligned}&\log \left(\mathrm {B} (\alpha ,\beta )\right)+(\alpha -1)(\psi (\beta )-\psi (\alpha ))\\+&(\alpha +\beta )\left(\psi (1-\alpha -\beta )-\psi (1-\beta )+{\frac {\pi \sin(\alpha \pi )}{\sin(\beta \pi )\sin((\alpha +\beta )\pi ))}}\right)\end{aligned}}$ where $\psi$ is the digamma function.
MGF	Does not exist
CF	${\frac {e^{-it}\Gamma (\alpha +\beta )}{\Gamma (\beta )}}G_{1,2}^{\,2,0}\!\left(\left.{\begin{matrix}\alpha +\beta \\\beta ,0\end{matrix}}\;\right|\,-it\right)$

Definitions

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

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

where B is the Beta function.

The cumulative distribution function is

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

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.

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

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

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

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

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

The cdf can also be written as

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

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

Alternative parameterization

The beta prime distribution may also be reparameterized in terms of its mean μ > 0 and precision ν > 0 parameters ( 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.

Generalization

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

• $p>0$ shape (real)
• $q>0$ scale (real)

having the probability density function:

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

with mean

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

and mode

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

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 $y\sim \beta '(\alpha ,\beta )$ and $x=qy^{1/p}$ for $q,p>0$, then $x\sim \beta '(\alpha ,\beta ,p,q)$.

Compound gamma distribution

The compound gamma distribution 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:

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

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

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 $r\sim G(\beta ,q)$ and $x\mid r\sim G(\alpha ,r)$, then $x\sim \beta '(\alpha ,\beta ,1,q)$. 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.

Properties

• If $X\sim \beta '(\alpha ,\beta )$ then ${\tfrac {1}{X}}\sim \beta '(\beta ,\alpha )$.
• If $Y\sim \beta '(\alpha ,\beta )$, and $X=qY^{1/p}$, then $X\sim \beta '(\alpha ,\beta ,p,q)$.
• If $X\sim \beta '(\alpha ,\beta ,p,q)$ then $kX\sim \beta '(\alpha ,\beta ,p,kq)$.
• $\beta '(\alpha ,\beta ,1,1)=\beta '(\alpha ,\beta )$

Related distributions

• If $X\sim {\textrm {Beta}}(\alpha ,\beta )$, then ${\frac {X}{1-X}}\sim \beta '(\alpha ,\beta )$. This property can be used to generate beta prime distributed variates.
• If $X\sim \beta '(\alpha ,\beta )$, then ${\frac {X}{1+X}}\sim {\textrm {Beta}}(\alpha ,\beta )$. This is a corollary from the property above.
• If $X\sim F(2\alpha ,2\beta )$ has an F-distribution, then ${\tfrac {\alpha }{\beta }}X\sim \beta '(\alpha ,\beta )$, or equivalently, $X\sim \beta '(\alpha ,\beta ,1,{\tfrac {\beta }{\alpha }})$.
• For gamma distribution parametrization I:
  • If $X_{k}\sim \Gamma (\alpha _{k},\theta _{k})$ are independent, then ${\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\theta _{1}}{\theta _{2}}})$. Note $\alpha _{1},\alpha _{2},{\tfrac {\theta _{1}}{\theta _{2}}}$ are all scale parameters for their respective distributions.
• For gamma distribution parametrization II:
  • If $X_{k}\sim \Gamma (\alpha _{k},\beta _{k})$ are independent, then ${\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\beta _{2}}{\beta _{1}}})$. The $\beta _{k}$ are rate parameters, while ${\tfrac {\beta _{2}}{\beta _{1}}}$ is a scale parameter.
  • If $\beta _{2}\sim \Gamma (\alpha _{1},\beta _{1})$ and $X_{2}\mid \beta _{2}\sim \Gamma (\alpha _{2},\beta _{2})$, then $X_{2}\sim \beta '(\alpha _{2},\alpha _{1},1,\beta _{1})$. The $\beta _{k}$ are rate parameters for the gamma distributions, but $\beta _{1}$ is the scale parameter for the beta prime.
• $\beta '(p,1,a,b)={\textrm {Dagum}}(p,a,b)$ the Dagum distribution
• $\beta '(1,p,a,b)={\textrm {SinghMaddala}}(p,a,b)$ the Singh–Maddala distribution.
• $\beta '(1,1,\gamma ,\sigma )={\textrm {LL}}(\gamma ,\sigma )$ 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 $x_{m}$ and shape parameter $\alpha$, then ${\dfrac {X}{x_{m}}}-1\sim \beta ^{\prime }(1,\alpha )$.
• If X has a Lomax distribution, also known as a Pareto Type II distribution, with shape parameter $\alpha$ and scale parameter $\lambda$, then ${\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )$.
• If X has a standard Pareto Type IV distribution with shape parameter $\alpha$ and inequality parameter $\gamma$, then $X^{\frac {1}{\gamma }}\sim \beta ^{\prime }(1,\alpha )$, or equivalently, $X\sim \beta ^{\prime }(1,\alpha ,{\tfrac {1}{\gamma }},1)$.
• The inverted Dirichlet distribution is a generalization of the beta prime distribution.
• If $X\sim \beta '(\alpha ,\beta )$, then $\ln X$ has a generalized logistic distribution. More generally, if $X\sim \beta '(\alpha ,\beta ,p,q)$, then $\ln X$ has a scaled and shifted generalized logistic distribution.
• If $X\sim \beta '\left({\frac {1}{2}},{\frac {1}{2}}\right)$, then $\pm {\sqrt {X}}$ follows a Cauchy distribution, which is equivalent to a student-t distribution with the degrees of freedom of 1.

References

• 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