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.

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


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

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 $$\beta'(\alpha,\beta)$$ 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


 * $$ \frac{x^\alpha \cdot {}_2F_1(\alpha, \alpha+\beta, \alpha+1, -x)}{\alpha \cdot B(\alpha,\beta)}$$

where $${}_2F_1$$ 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:


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

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


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

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 $$y\sim\beta'(\alpha,\beta)$$ 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 $$kX \sim \beta'(\alpha,\beta,p,kq) $$.
 * $$\beta'(\alpha,\beta,1,1) = \beta'(\alpha,\beta) $$
 * If $$ X_1 \sim \beta'(\alpha,\beta) $$ and $$ X_2 \sim \beta'(\alpha,\beta) $$ two iid variables, then $$ Y=X_1+X_2 \sim \beta'(\gamma,\delta) $$ 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. $$\forall i, 1\leq i\leq n, X_i \sim \beta'(\alpha,\beta)$$, then the sum $$ S=X_1+...+X_n \sim \beta'(\gamma,\delta) $$ 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 $$\tfrac{\alpha}{\beta} X \sim \beta'(\alpha,\beta)$$, 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 $$X \sim \beta'(\alpha,\beta)$$ then $$\frac{X}{1+X} \sim \textrm{Beta}(\alpha,\beta) $$.
 * 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.