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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 has a beta distribution, then the odds has a beta prime distribution.

Definitions

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

where B is the Beta function.

The cumulative distribution function is

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 is . Its mean is if (if the mean is infinite, in other words it has no well defined mean) and its variance is if .

For , the k-th moment is given by

For with this simplifies to

The cdf can also be written as

where is the Gauss's hypergeometric function <sub>2</sub>F<sub>1</sub>&nbsp;.

Alternative parameterization

The beta prime distribution may also be reparameterized in terms of its mean μ&nbsp;>&nbsp;0 and precision ν&nbsp;>&nbsp;0 parameters ( p.&nbsp;36).

Consider the parameterization μ =&nbsp;α/(β&nbsp;−&nbsp;1) and ν =&nbsp;β&nbsp;−&nbsp;2, i.e., α =&nbsp;μ(1&nbsp;+&nbsp;ν) and β =&nbsp;2&nbsp;+&nbsp;ν. Under this parameterization E[Y]&nbsp;=&nbsp;μ and Var[Y]&nbsp;=&nbsp;μ(1&nbsp;+&nbsp;μ)/ν.

Generalization

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

having the probability density function:

with mean

and mode

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 and for , then .

Compound gamma distribution

The compound gamma distribution is the generalization of the beta prime when the scale parameter, q is added, but where p&nbsp;=&nbsp;1. It is so named because it is formed by compounding two gamma distributions:

where is the gamma pdf with shape and inverse scale .

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<sup>2</sup>.

Another way to express the compounding is if and , then . 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 then .
  • If , and , then .
  • If then .

Related distributions

  • If , then . This property can be used to generate beta prime distributed variates.
  • If , then . This is a corollary from the property above.
  • If has an F-distribution, then , or equivalently, .
  • For gamma distribution parametrization I:
  • If are independent, then . Note are all scale parameters for their respective distributions.
  • For gamma distribution parametrization II:
  • If are independent, then . The are rate parameters, while is a scale parameter.
  • If and , then . The are rate parameters for the gamma distributions, but is the scale parameter for the beta prime.
  • the Dagum distribution
  • the Singh–Maddala distribution.
  • 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 and shape parameter , then .
  • If X has a Lomax distribution, also known as a Pareto Type II distribution, with shape parameter and scale parameter , then .
  • If X has a standard Pareto Type IV distribution with shape parameter and inequality parameter , then , or equivalently, .
  • The inverted Dirichlet distribution is a generalization of the beta prime distribution.
  • If , then has a generalized logistic distribution. More generally, if , then has a scaled and shifted generalized logistic distribution.
  • If , then 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.