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Definitions

Beta prime distribution is defined for x > 0 {\displaystyle x>0} with two parameters α and β, having the probability density function:

f ( x ) = x α − 1 ( 1 + x ) − α − β B ( α , β ) {\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{\mathrm {B} (\alpha ,\beta )}}}

where B is the Beta function.

The cumulative distribution function is

F ( x ; α , β ) = I x 1 + x ( α , β ) , {\displaystyle F(x;\alpha ,\beta )=I_{\frac {x}{1+x}}\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.²

The mode of a variate X distributed as β ′ ( α , β ) {\displaystyle \beta '(\alpha ,\beta )} is X ^ = α − 1 β + 1 {\displaystyle {\hat {X}}={\frac {\alpha -1}{\beta +1}}} . Its mean is α β − 1 {\displaystyle {\frac {\alpha }{\beta -1}}} if β > 1 {\displaystyle \beta >1} (if β ≤ 1 {\displaystyle \beta \leq 1} the mean is infinite, in other words it has no well defined mean) and its variance is α ( α + β − 1 ) ( β − 2 ) ( β − 1 ) 2 {\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}} if β > 2 {\displaystyle \beta >2} .

For − α < k < β {\displaystyle -\alpha <k<\beta } , the k-th moment E [ X k ] {\displaystyle E[X^{k}]} is given by

E [ X k ] = B ( α + k , β − k ) B ( α , β ) . {\displaystyle E[X^{k}]={\frac {\mathrm {B} (\alpha +k,\beta -k)}{\mathrm {B} (\alpha ,\beta )}}.}

For k ∈ N {\displaystyle k\in \mathbb {N} } with k < β , {\displaystyle k<\beta ,} this simplifies to

E [ X k ] = ∏ i = 1 k α + i − 1 β − i . {\displaystyle E[X^{k}]=\prod _{i=1}^{k}{\frac {\alpha +i-1}{\beta -i}}.}

The cdf can also be written as

x α ⋅ 2 F 1 ( α , α + β , α + 1 , − x ) α ⋅ B ( α , β ) {\displaystyle {\frac {x^{\alpha }\cdot {}_{2}F_{1}(\alpha ,\alpha +\beta ,\alpha +1,-x)}{\alpha \cdot \mathrm {B} (\alpha ,\beta )}}}

where 2 F 1 {\displaystyle {}_{2}F_{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 (³ 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 β ′ ( α , β , p , q ) {\displaystyle \beta '(\alpha ,\beta ,p,q)} :

• p > 0 {\displaystyle p>0} shape (real)
• q > 0 {\displaystyle q>0} scale (real)

having the probability density function:

f ( x ; α , β , p , q ) = p ( x q ) α p − 1 ( 1 + ( x q ) p ) − α − β q B ( α , β ) {\displaystyle 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 }}{q\mathrm {B} (\alpha ,\beta )}}}

with mean

q Γ ( α + 1 p ) Γ ( β − 1 p ) Γ ( α ) Γ ( β ) if  β p > 1 {\displaystyle {\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 ( α p − 1 β p + 1 ) 1 p if  α p ≥ 1 {\displaystyle q\left({\frac {\alpha p-1}{\beta p+1}}\right)^{\tfrac {1}{p}}\quad {\text{if }}\alpha p\geq 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 ∼ β ′ ( α , β ) {\displaystyle y\sim \beta '(\alpha ,\beta )} and x = q y 1 / p {\displaystyle x=qy^{1/p}} for q , p > 0 {\displaystyle q,p>0} , then x ∼ β ′ ( α , β , p , q ) {\displaystyle x\sim \beta '(\alpha ,\beta ,p,q)} .

Compound gamma distribution

The compound gamma distribution⁴ 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:

β ′ ( x ; α , β , 1 , q ) = ∫ 0 ∞ G ( x ; α , r ) G ( r ; β , q ) d r {\displaystyle \beta '(x;\alpha ,\beta ,1,q)=\int _{0}^{\infty }G(x;\alpha ,r)G(r;\beta ,q)\;dr}

where G ( x ; a , b ) {\displaystyle G(x;a,b)} is the gamma pdf with shape a {\displaystyle a} and inverse scale b {\displaystyle 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 ∼ G ( β , q ) {\displaystyle r\sim G(\beta ,q)} and x ∣ r ∼ G ( α , r ) {\displaystyle x\mid r\sim G(\alpha ,r)} , then x ∼ β ′ ( α , β , 1 , q ) {\displaystyle 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 ∼ β ′ ( α , β ) {\displaystyle X\sim \beta '(\alpha ,\beta )} then 1 X ∼ β ′ ( β , α ) {\displaystyle {\tfrac {1}{X}}\sim \beta '(\beta ,\alpha )} .
• If Y ∼ β ′ ( α , β ) {\displaystyle Y\sim \beta '(\alpha ,\beta )} , and X = q Y 1 / p {\displaystyle X=qY^{1/p}} , then X ∼ β ′ ( α , β , p , q ) {\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)} .
• If X ∼ β ′ ( α , β , p , q ) {\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)} then k X ∼ β ′ ( α , β , p , k q ) {\displaystyle kX\sim \beta '(\alpha ,\beta ,p,kq)} .
• β ′ ( α , β , 1 , 1 ) = β ′ ( α , β ) {\displaystyle \beta '(\alpha ,\beta ,1,1)=\beta '(\alpha ,\beta )}

Related distributions

• If X ∼ Beta ( α , β ) {\displaystyle X\sim {\textrm {Beta}}(\alpha ,\beta )} , then X 1 − X ∼ β ′ ( α , β ) {\displaystyle {\frac {X}{1-X}}\sim \beta '(\alpha ,\beta )} . This property can be used to generate beta prime distributed variates.
• If X ∼ β ′ ( α , β ) {\displaystyle X\sim \beta '(\alpha ,\beta )} , then X 1 + X ∼ Beta ( α , β ) {\displaystyle {\frac {X}{1+X}}\sim {\textrm {Beta}}(\alpha ,\beta )} . This is a corollary from the property above.
• If X ∼ F ( 2 α , 2 β ) {\displaystyle X\sim F(2\alpha ,2\beta )} has an F-distribution, then α β X ∼ β ′ ( α , β ) {\displaystyle {\tfrac {\alpha }{\beta }}X\sim \beta '(\alpha ,\beta )} , or equivalently, X ∼ β ′ ( α , β , 1 , β α ) {\displaystyle X\sim \beta '(\alpha ,\beta ,1,{\tfrac {\beta }{\alpha }})} .
• For gamma distribution parametrization I:
  • If X k ∼ Γ ( α k , θ k ) {\displaystyle X_{k}\sim \Gamma (\alpha _{k},\theta _{k})} are independent, then X 1 X 2 ∼ β ′ ( α 1 , α 2 , 1 , θ 1 θ 2 ) {\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\theta _{1}}{\theta _{2}}})} . Note θ 1 , θ 2 , θ 1 θ 2 {\displaystyle \theta _{1},\theta _{2},{\tfrac {\theta _{1}}{\theta _{2}}}} are all scale parameters for their respective distributions.
• For gamma distribution parametrization II:
  • If X k ∼ Γ ( α k , β k ) {\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})} are independent, then X 1 X 2 ∼ β ′ ( α 1 , α 2 , 1 , β 2 β 1 ) {\displaystyle {\tfrac {X_{1}}{X_{2}}}\sim \beta '(\alpha _{1},\alpha _{2},1,{\tfrac {\beta _{2}}{\beta _{1}}})} . The β k {\displaystyle \beta _{k}} are rate parameters, while β 2 β 1 {\displaystyle {\tfrac {\beta _{2}}{\beta _{1}}}} is a scale parameter.
  • If β 2 ∼ Γ ( α 1 , β 1 ) {\displaystyle \beta _{2}\sim \Gamma (\alpha _{1},\beta _{1})} and X 2 ∣ β 2 ∼ Γ ( α 2 , β 2 ) {\displaystyle X_{2}\mid \beta _{2}\sim \Gamma (\alpha _{2},\beta _{2})} , then X 2 ∼ β ′ ( α 2 , α 1 , 1 , β 1 ) {\displaystyle X_{2}\sim \beta '(\alpha _{2},\alpha _{1},1,\beta _{1})} . The β k {\displaystyle \beta _{k}} are rate parameters for the gamma distributions, but β 1 {\displaystyle \beta _{1}} is the scale parameter for the beta prime.
• β ′ ( p , 1 , a , b ) = Dagum ( p , a , b ) {\displaystyle \beta '(p,1,a,b)={\textrm {Dagum}}(p,a,b)} the Dagum distribution
• β ′ ( 1 , p , a , b ) = SinghMaddala ( p , a , b ) {\displaystyle \beta '(1,p,a,b)={\textrm {SinghMaddala}}(p,a,b)} the Singh–Maddala distribution.
• β ′ ( 1 , 1 , γ , σ ) = LL ( γ , σ ) {\displaystyle \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 {\displaystyle x_{m}} and shape parameter α {\displaystyle \alpha } , then X x m − 1 ∼ β ′ ( 1 , α ) {\displaystyle {\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 α {\displaystyle \alpha } and scale parameter λ {\displaystyle \lambda } , then X λ ∼ β ′ ( 1 , α ) {\displaystyle {\frac {X}{\lambda }}\sim \beta ^{\prime }(1,\alpha )} .
• If X has a standard Pareto Type IV distribution with shape parameter α {\displaystyle \alpha } and inequality parameter γ {\displaystyle \gamma } , then X 1 γ ∼ β ′ ( 1 , α ) {\displaystyle X^{\frac {1}{\gamma }}\sim \beta ^{\prime }(1,\alpha )} , or equivalently, X ∼ β ′ ( 1 , α , 1 γ , 1 ) {\displaystyle X\sim \beta ^{\prime }(1,\alpha ,{\tfrac {1}{\gamma }},1)} .
• The inverted Dirichlet distribution is a generalization of the beta prime distribution.
• If X ∼ β ′ ( α , β ) {\displaystyle X\sim \beta '(\alpha ,\beta )} , then ln ⁡ X {\displaystyle \ln X} has a generalized logistic distribution. More generally, if X ∼ β ′ ( α , β , p , q ) {\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)} , then ln ⁡ X {\displaystyle \ln X} has a scaled and shifted generalized logistic distribution.
• If X ∼ β ′ ( 1 2 , 1 2 ) {\displaystyle X\sim \beta '\left({\frac {1}{2}},{\frac {1}{2}}\right)} , then ± X {\displaystyle \pm {\sqrt {X}}} follows a Cauchy distribution, which is equivalent to a student-t distribution with the degrees of freedom of 1.

Notes

• Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley. ISBN 0-471-58494-0
• 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, doi:10.1007/s40300-021-00203-y, S2CID 233534544

• MathWorld article

References

1. Johnson et al (1995), p 248 ↩

2. Johnson et al (1995), p 248 ↩

3. 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. doi:10.1007/s40300-021-00203-y. S2CID 233534544. /wiki/Doi_(identifier) ↩

4. Dubey, Satya D. (December 1970). "Compound gamma, beta and F distributions". Metrika. 16: 27–31. doi:10.1007/BF02613934. S2CID 123366328. /wiki/Doi_(identifier) ↩