Beta prime distribution

Short description: Probability distribution

Beta prime

Probability density function
Cumulative distribution function
Parameters	$\alpha >0$ shape (real) $\beta >0$ shape (real)
Support	$x\in [0,\mathrm{\infty })$
PDF	$f(x)=\frac{x^{\alpha -1}(1+x{)}^{-\alpha -\beta }}{B(\alpha ,\beta )}$
CDF	$I_{\frac{x}{1+x}(\alpha ,\beta )}$ where $I_x(\alpha ,\beta )$ is the incomplete beta function
Mean	$\frac{\alpha }{\beta -1}\text{ if }\beta >1$
Mode	$\frac{\alpha -1}{\beta +1}\text{ if }\alpha \ge 1\text{, 0 otherwise}$
Variance	$\frac{\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1{)}^2}\text{ if }\beta >2$
Skewness	$\frac{2(2\alpha +\beta -1)}{\beta -3}\sqrt{\frac{\beta -2}{\alpha (\alpha +\beta -1)}}\text{ if }\beta >3$
MGF	Does not exist
CF	$\frac{e^{-it}\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }(\beta )}{G}_{1,2}^{2,0}(\begin{array}{c}\alpha +\beta \\ \beta ,0\end{array}|-it)$

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.

Definitions

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

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

where B is the Beta function.

The cumulative distribution function is

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

where I is the regularized incomplete beta function.

The expected 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 ${\beta }^{\prime }(\alpha ,\beta )$ is $\hat{X}=\frac{\alpha -1}{\beta +1}$. Its mean is $\frac{\alpha }{\beta -1}$ if $\beta >1$ (if $\beta \le 1$ 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 $k\in \mathbb{\mathbb{N}}$ 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 .

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 ${\beta }^{\prime }(\alpha ,\beta ,p,q)$:

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

having the probability density function:

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

with mean

$\frac{q\mathrm{\Gamma }(\alpha +\frac{1}{p})\mathrm{\Gamma }(\beta -\frac{1}{p})}{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\beta )}\text{if }\beta p>1$

and mode

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

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

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:

${\beta }^{\prime }(x;\alpha ,\beta ,1,q)=\underset{0}{\overset{\mathrm{\infty }}{\int }}G(x;\alpha ,r)G(r;\beta ,q)dr$

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 $x\sim {\beta }^{\prime }(\alpha ,\beta ,1,q)$. (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 $X\sim {\beta }^{\prime }(\alpha ,\beta )$ then $\frac{1}{X}\sim {\beta }^{\prime }(\beta ,\alpha )$.
• If $Y\sim {\beta }^{\prime }(\alpha ,\beta )$, and $X=q{Y}^{1/p}$, then $X\sim {\beta }^{\prime }(\alpha ,\beta ,p,q)$.
• If $X\sim {\beta }^{\prime }(\alpha ,\beta ,p,q)$ then $kX\sim {\beta }^{\prime }(\alpha ,\beta ,p,kq)$.
• ${\beta }^{\prime }(\alpha ,\beta ,1,1)={\beta }^{\prime }(\alpha ,\beta )$
• If $X_1\sim {\beta }^{\prime }(\alpha ,\beta )$ and $X_2\sim {\beta }^{\prime }(\alpha ,\beta )$ two iid variables, then $Y=X_1+X_2\sim {\beta }^{\prime }(\gamma ,\delta )$ 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. $\mathrm{\forall }i,1\le i\le n,X_i\sim {\beta }^{\prime }(\alpha ,\beta )$, then the sum $S=X_1+...+X_n\sim {\beta }^{\prime }(\gamma ,\delta )$ with $\gamma =\frac{n\alpha (\alpha +{\beta }^2-2\beta +n\alpha \beta -2n\alpha +1)}{(\beta -1)(\alpha +\beta -1)}$ and $\delta =\frac{2\alpha +{\beta }^2-\beta +n\alpha \beta -2n\alpha }{\alpha +\beta -1}$.

Related distributions

• If $X\sim F(2\alpha ,2\beta )$ has an F-distribution, then $\frac{\alpha }{\beta }X\sim {\beta }^{\prime }(\alpha ,\beta )$, or equivalently, $X\sim {\beta }^{\prime }(\alpha ,\beta ,1,\frac{\beta }{\alpha })$.
• If $X\sim \text{<mrow data-mjx-texclass="ORD">Beta</mrow>}(\alpha ,\beta )$ then $\frac{X}{1-X}\sim {\beta }^{\prime }(\alpha ,\beta )$.
• If $X\sim {\beta }^{\prime }(\alpha ,\beta )$ then $\frac{X}{1+X}\sim \text{<mrow data-mjx-texclass="ORD">Beta</mrow>}(\alpha ,\beta )$.
• For gamma distribution parametrization I:
  • If $X_k\sim \mathrm{\Gamma }({\alpha }_k,{\theta }_k)$ are independent, then $\frac{X_1}{X_2}\sim {\beta }^{\prime }({\alpha }_1,{\alpha }_2,1,\frac{{\theta }_1}{{\theta }_2})$. Note ${\theta }_1,{\theta }_2,\frac{{\theta }_1}{{\theta }_2}$ are all scale parameters for their respective distributions.
• For gamma distribution parametrization II:
  • If $X_k\sim \mathrm{\Gamma }({\alpha }_k,{\beta }_k)$ are independent, then $\frac{X_1}{X_2}\sim {\beta }^{\prime }({\alpha }_1,{\alpha }_2,1,\frac{{\beta }_2}{{\beta }_1})$. The ${\beta }_k$ are rate parameters, while $\frac{{\beta }_2}{{\beta }_1}$ is a scale parameter.
  • If ${\beta }_2\sim \mathrm{\Gamma }({\alpha }_1,{\beta }_1)$ and $X_2\mid {\beta }_2\sim \mathrm{\Gamma }({\alpha }_2,{\beta }_2)$, then $X_2\sim {\beta }^{\prime }({\alpha }_2,{\alpha }_1,1,{\beta }_1)$. The ${\beta }_k$ are rate parameters for the gamma distributions, but ${\beta }_1$ is the scale parameter for the beta prime.
• ${\beta }^{\prime }(p,1,a,b)=\text{<mrow data-mjx-texclass="ORD">Dagum</mrow>}(p,a,b)$ the Dagum distribution
• ${\beta }^{\prime }(1,p,a,b)=\text{<mrow data-mjx-texclass="ORD">SinghMaddala</mrow>}(p,a,b)$ the Singh–Maddala distribution.
• ${\beta }^{\prime }(1,1,\gamma ,\sigma )=\text{<mrow data-mjx-texclass="ORD">LL</mrow>}(\gamma ,\sigma )$ 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 $\frac{X}{x_m}-1\sim {\beta }^{\prime }(1,\alpha )$.
• 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, $X\sim {\beta }^{\prime }(1,\alpha ,\frac{1}{\gamma },1)$.
• The inverted Dirichlet distribution is a generalization of the beta prime distribution.
• If $X\sim {\beta }^{\prime }(\alpha ,\beta )$, then $\ln X$ has a generalized logistic distribution. More generally, if $X\sim {\beta }^{\prime }(\alpha ,\beta ,p,q)$, then $\ln X$ has a scaled and shifted generalized logistic distribution.

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 

• MathWorld article