Beta prime distribution explained

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. If

p\in[ 0 , 1 ] has a beta distribution, then the odds (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) = (x ^{\alpha-1} ( 1+ x) ^{-\alpha})/(B(\alpha,\beta))

where B is the Beta function.

The cumulative distribution function is

F(x; \alpha,\beta) = I _{(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 (\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.^[1]

The mode of a variate X distributed as

\beta'(\alpha,\beta) is \hat{X} = (\alpha -1)/(\beta +1) .Its mean is (\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 (\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 byE[X ^{k] = (B(\alpha + k,\beta - k))/(B(\alpha,\beta))} .

For

k\in N with k <\beta, this simplifies toE[X ^{k} _{i =1} (\alpha + i -1)/(\beta - i) .

The cdf can also be written as

(x ^{\alpha} ⋅ {)/(_{2F}) _{1(\alpha,} \alpha + \beta, \alpha +1 , - x)}{\alpha ⋅ 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 (^[2] p. 36).

Consider the parameterization μ = α/(β-1) and ν = β- 2, i.e., α = μ(1 + ν) andβ = 2 + ν. Under this parameterizationE[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) = (p \left( (x)/(q) \right) ^{\alpha} \left( 1+ \left( (x)/(q) \right) ^{p\right)} ^{-\alpha})/(qB(\alpha,\beta))

with mean

(q\Gamma\left(\alpha + \tfrac 1 p\right)\Gamma(\beta - \tfrac 1 p))/(\Gamma(\alpha)\Gamma(\beta)) if \beta p> 1

and mode

q \left({ (\alpha p -1)/(\beta p +1) }\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 ^{1/p} for q,p> 0 , then x\sim\beta'(\alpha,\beta,p,q) .

Compound gamma distribution

The compound gamma distribution^[3] 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 ^{inf ty} _{0} 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 = (2 \alpha(\alpha + \beta ^{2} -2 \beta +2 \alpha\beta -4 \alpha +1 ))/((\beta -1 )(\alpha + \beta -1 )) and \delta = (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 = (n\alpha(\alpha + \beta ^{2} -2 \beta + n\alpha\beta -2 n\alpha +1 ))/((\beta -1 )(\alpha + \beta -1 )) and \delta = (2 \alpha + \beta ^{2-\beta} + n\alpha \beta -2 n\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 rm{Beta}(\alpha,\beta) then (X)/(1- X) \sim \beta'(\alpha,\beta) .

• If

X \sim \beta'(\alpha,\beta) then (X)/(1+ X) \sim rm{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) = rm{Dagum}(p,a,b) the Dagum distribution\beta'( 1 ,p,a,b) = rm{SinghMaddala}(p,a,b) the Singh–Maddala distribution.\beta'( 1 , 1 ,\gamma,\sigma) = rm{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 ^{\prime( 1 ,\alpha)} _{m} -1 \sim\beta} .

• If X has a Lomax distribution, also known as a Pareto Type II distribution, with shape parameter

\alpha and scale parameter λ , then (X)/(λ) \sim \beta ^{\prime(1,\alpha)} .

• If X has a standard Pareto Type IV distribution with shape parameter

\alpha and inequality parameter \gamma , then X ^{(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.

References

• Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley.
  • MathWorld article

Notes and References

1. Johnson et al (1995), p 248
2. 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 . 10.1007/s40300-021-00203-y. 233534544 .
3. Dubey. Satya D.. Compound gamma, beta and F distributions. Metrika. December 1970. 16. 27–31. 10.1007/BF02613934. 123366328.