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[1]) 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 a 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[1].

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 .

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

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

and mode

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

Note that if p=q=1 then the generalized beta prime distribution reduces to the standard beta prime distribution

Compound gamma distribution

The compound gamma distribution[2] 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:

\( \beta'(x;\alpha,\beta,1,q) = \int_0^\infty G(x;\alpha,p)G(p;\beta,q) \; dp \)

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

If \( X \sim F(\alpha,\beta)\, \) then \( \tfrac{\alpha}{\beta} X \sim \beta^{'}(\tfrac{\alpha}{2},\tfrac{\beta}{2})\, \)
If \( X \sim \textrm{Beta}(\alpha,\beta)\, \) then \( \frac{X}{1-X} \sim \beta^{'}(\alpha,\beta)\, \)
If \( X \sim \Gamma(\alpha,1)\, \) and \( Y \sim \Gamma(\beta,1)\,, \) then \( \frac{X}{Y} \sim \beta^{'}(\alpha,\beta). \)
\( \beta^{'}(p,1,a,b) = \textrm{Dagum}(p,a,b)\, \) 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
Beta prime distribution is a special case of the type 6 Pearson distribution
Pareto distribution type II is related to Beta prime distribution
Pareto distribution type IV is related to Beta prime distribution

Notes

^ a b Johnson et al (1995), p248
^ Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". Metrika 16: 27–31. doi:10.1007/BF02613934.

References

Jonhnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0

MathWorld article

Mathematics Encyclopedia