Generalized Beta Prime Distribution

In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kindJohnson et al (1995), p 248) is an absolutely continuous probability distribution. If p\in ,1/math> has a beta distribution, then the odds $\frac$ 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:

: $f(x) = \frac$

where ''B'' is the Beta function.

The cumulative distribution function is

: $F(x; \alpha,\beta)=I_\left(\alpha, \beta \right) ,$

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 $\hat = \frac$.
Its mean is $\frac$ if $\beta>1$ (if $\beta \leq 1$ the mean is infinite, in other words it has no well defined mean) and its variance is $\frac$ if $\beta>2$.

For -\alpha , the ''k''-th moment E ^k is given by

: E ^k\frac.

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

: E ^k\prod_^k \frac.

The cdf can also be written as

:$\frac$

where $_2F_1$ is the Gauss's hypergeometric function ₂''F''₁ .

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''nbsp;= ''μ'' and Var nbsp;= ''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:

: $f(x;\alpha,\beta,p,q) = \frac$

with mean

: $\frac \quad \text \beta p>1$

and mode

: $q \left(\right)^\tfrac \quad \text \alpha p\ge 1$

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 $x=qy^$ 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 ''q''².

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 $\tfrac \sim \beta'(\beta,\alpha)$.
*If $Y\sim\beta'(\alpha,\beta)$, and $X=qY^$, 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 $X \sim \textrm(\alpha,\beta)$, then $\frac \sim \beta'(\alpha,\beta)$. This property can be used to generate beta prime distributed variates.
* If $X \sim \beta'(\alpha,\beta)$, then $\frac \sim \textrm(\alpha,\beta)$. This is a corollary from the property above.
* If $X \sim F(2\alpha,2\beta)$ has an ''F''-distribution, then $\tfrac X \sim \beta'(\alpha,\beta)$, or equivalently, $X\sim\beta'(\alpha,\beta , 1 , \tfrac)$.
* For gamma distribution parametrization I:
** If $X_k \sim \Gamma(\alpha_k,\theta_k)$ are independent, then $\tfrac \sim \beta'(\alpha_1,\alpha_2,1,\tfrac)$. Note $\theta_1,\theta_2,\tfrac$ 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 $\tfrac \sim \beta'(\alpha_1,\alpha_2,1,\tfrac)$. The $\beta_k$ are rate parameters, while $\tfrac$ 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.
* $\beta'(p,1,a,b) = \textrm(p,a,b)$ the Dagum distribution
* $\beta'(1,p,a,b) = \textrm(p,a,b)$ the Singh–Maddala distribution.
* $\beta'(1,1,\gamma,\sigma) = \textrm(\gamma,\sigma)$ 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 $\dfrac-1\sim\beta^\prime(1,\alpha)$.
* If ''X'' has a Lomax distribution, also known as a Pareto Type II distribution, with shape parameter $\alpha$ and scale parameter $\lambda$, then $\frac\sim \beta^\prime(1,\alpha)$.
* If ''X'' has a standard Pareto Type IV distribution with shape parameter $\alpha$ and inequality parameter $\gamma$, then $X^ \sim \beta^\prime(1,\alpha)$, or equivalently, $X \sim \beta^\prime(1,\alpha,\tfrac,1)$.
* 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 $X\sim\beta'\left(\frac,\frac\right)$, then $\pm\sqrt$ follows a Cauchy distribution, which is equivalent to a student-t distribution with the degrees of freedom of 1.

Notes

References

* Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). ''Continuous Univariate Distributions'', Volume 2 (2nd Edition), Wiley.
*

MathWorld article

{{ProbDistributions, continuous-semi-infinite

Continuous distributions
Compound probability distributions