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

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

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.

The expectation value, variance, and other details of the distribution are given in the sidebox; for $\beta >4$, the excess kurtosis is

$\gamma _{2}=6{\frac {\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}}.$

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

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

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

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

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

Generalization

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

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

having the probability density function:

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

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

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 distribution with shape a and inverse scale b. This relationship can be used to generate random variables with a compound gamma, or beta prime distribution.

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.

Properties

• If $X\sim \beta '(\alpha ,\beta )$ then ${\tfrac {1}{X}}\sim \beta '(\beta ,\alpha )$.
• 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 and properties

• If $X\sim F(2\alpha ,2\beta )$ has an F-distribution, then ${\tfrac {\alpha }{\beta }}X\sim \beta '(\alpha ,\beta )$, or equivalently, $X\sim \beta '(\alpha ,\beta ,1,{\tfrac {\beta }{\alpha }})$.
• If $X\sim {\textrm {Beta}}(\alpha ,\beta )$ then ${\frac {X}{1-X}}\sim \beta '(\alpha ,\beta )$.
• If $X\sim \Gamma (\alpha ,1)$ and $Y\sim \Gamma (\beta ,1)$ are independent, then ${\frac {X}{Y}}\sim \beta '(\alpha ,\beta )$.
• Parametrization 1: If $X_{k}\sim \Gamma (\alpha _{k},\theta _{k})$ are independent, then ${\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\theta _{1}}{\theta _{2}}})$.
• Parametrization 2: If $X_{k}\sim \Gamma (\alpha _{k},\beta _{k})$ are independent, then ${\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\beta _{2}}{\beta _{1}}})$.
• $\beta '(p,1,a,b)={\textrm {Dagum}}(p,a,b)$ 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 $X-x_{m}\sim \beta ^{\prime }(1,\alpha )$.
• If X has a Lomax distribution, also known as a Pareto Type II distribution, with shape parameter $\alpha$ 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.

References

• Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley. Template:ISBN
• MathWorld article