Probability distribution
Beta prime Probability density function
Cumulative distribution function
Parameters α > 0 {\displaystyle \alpha >0} shape (real ) β > 0 {\displaystyle \beta >0} Support x ∈ [ 0 , ∞ ) {\displaystyle x\in [0,\infty )\!} PDF 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 )}}\!} CDF I x 1 + x ( α , β ) {\displaystyle I_{{\frac {x}{1+x}}(\alpha ,\beta )}} I x ( α , β ) {\displaystyle I_{x}(\alpha ,\beta )} Mean α β − 1 {\displaystyle {\frac {\alpha }{\beta -1}}} β > 1 {\displaystyle \beta >1} Mode α − 1 β + 1 if α ≥ 1 , 0 otherwise {\displaystyle {\frac {\alpha -1}{\beta +1}}{\text{ if }}\alpha \geq 1{\text{, 0 otherwise}}\!} Variance α ( α + β − 1 ) ( β − 2 ) ( β − 1 ) 2 {\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}} β > 2 {\displaystyle \beta >2} Skewness 2 ( 2 α + β − 1 ) β − 3 β − 2 α ( α + β − 1 ) {\displaystyle {\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}} β > 3 {\displaystyle \beta >3} Excess kurtosis 6 α ( α + β − 1 ) ( 5 β − 11 ) + ( β − 1 ) 2 ( β − 2 ) α ( α + β − 1 ) ( β − 3 ) ( β − 4 ) {\displaystyle 6{\frac {\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}}} β > 4 {\displaystyle \beta >4} Entropy log ( B ( α , β ) ) + ( α − 1 ) ( ψ ( β ) − ψ ( α ) ) + ( α + β ) ( ψ ( 1 − α − β ) − ψ ( 1 − β ) + π sin ( α π ) sin ( β π ) sin ( ( α + β ) π ) ) ) {\displaystyle {\begin{aligned}&\log \left(\mathrm {B} (\alpha ,\beta )\right)+(\alpha -1)(\psi (\beta )-\psi (\alpha ))\\+&(\alpha +\beta )\left(\psi (1-\alpha -\beta )-\psi (1-\beta )+{\frac {\pi \sin(\alpha \pi )}{\sin(\beta \pi )\sin((\alpha +\beta )\pi ))}}\right)\end{aligned}}} ψ {\displaystyle \psi } digamma function . MGF Does not exist CF e − i t Γ ( α + β ) Γ ( β ) G 1 , 2 2 , 0 ( α + β β , 0 | − i t ) {\displaystyle {\frac {e^{-it}\Gamma (\alpha +\beta )}{\Gamma (\beta )}}G_{1,2}^{\,2,0}\!\left(\left.{\begin{matrix}\alpha +\beta \\\beta ,0\end{matrix}}\;\right|\,-it\right)}
In probability theory and statistics , the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind [ 1] absolutely continuous probability distribution . If p ∈ [ 0 , 1 ] {\displaystyle p\in [0,1]} beta distribution , then the odds p 1 − p {\displaystyle {\frac {p}{1-p}}}
Beta prime distribution is defined for x > 0 {\displaystyle x>0} 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.[ 1]
The mode of a variate X distributed as β ′ ( α , β ) {\displaystyle \beta '(\alpha ,\beta )} X ^ = α − 1 β + 1 {\displaystyle {\hat {X}}={\frac {\alpha -1}{\beta +1}}} α β − 1 {\displaystyle {\frac {\alpha }{\beta -1}}} β > 1 {\displaystyle \beta >1} β ≤ 1 {\displaystyle \beta \leq 1} α ( α + β − 1 ) ( β − 2 ) ( β − 1 ) 2 {\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}} β > 2 {\displaystyle \beta >2}
For − α < k < β {\displaystyle -\alpha <k<\beta } k -th moment E [ X k ] {\displaystyle E[X^{k}]}
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} } k < β , {\displaystyle k<\beta ,}
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}} hypergeometric function 2 F 1 .
Alternative parameterization [ edit ] The beta prime distribution may also be reparameterized in terms of its mean μ > 0 and precision ν > 0 parameters ([ 2]
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 .
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} 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 )} x = q y 1 / p {\displaystyle x=qy^{1/p}} q , p > 0 {\displaystyle q,p>0} x ∼ β ′ ( α , β , p , q ) {\displaystyle x\sim \beta '(\alpha ,\beta ,p,q)}
Compound gamma distribution [ edit ] The compound gamma distribution [ 3] 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)} a {\displaystyle a} 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 q 2 .
Another way to express the compounding is if r ∼ G ( β , q ) {\displaystyle r\sim G(\beta ,q)} x ∣ r ∼ G ( α , r ) {\displaystyle x\mid r\sim G(\alpha ,r)} x ∼ β ′ ( α , β , 1 , q ) {\displaystyle x\sim \beta '(\alpha ,\beta ,1,q)}
If X ∼ β ′ ( α , β ) {\displaystyle X\sim \beta '(\alpha ,\beta )} 1 X ∼ β ′ ( β , α ) {\displaystyle {\tfrac {1}{X}}\sim \beta '(\beta ,\alpha )} If Y ∼ β ′ ( α , β ) {\displaystyle Y\sim \beta '(\alpha ,\beta )} X = q Y 1 / p {\displaystyle X=qY^{1/p}} 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)} 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 )} If X ∼ Beta ( α , β ) {\displaystyle X\sim {\textrm {Beta}}(\alpha ,\beta )} X 1 − X ∼ β ′ ( α , β ) {\displaystyle {\frac {X}{1-X}}\sim \beta '(\alpha ,\beta )} If X ∼ β ′ ( α , β ) {\displaystyle X\sim \beta '(\alpha ,\beta )} X 1 + X ∼ Beta ( α , β ) {\displaystyle {\frac {X}{1+X}}\sim {\textrm {Beta}}(\alpha ,\beta )} If X ∼ F ( 2 α , 2 β ) {\displaystyle X\sim F(2\alpha ,2\beta )} F -distribution α β X ∼ β ′ ( α , β ) {\displaystyle {\tfrac {\alpha }{\beta }}X\sim \beta '(\alpha ,\beta )} 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})} 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}}})} θ 1 , θ 2 , θ 1 θ 2 {\displaystyle \theta _{1},\theta _{2},{\tfrac {\theta _{1}}{\theta _{2}}}} For gamma distribution parametrization II: If X k ∼ Γ ( α k , β k ) {\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})} 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}}})} β k {\displaystyle \beta _{k}} β 2 β 1 {\displaystyle {\tfrac {\beta _{2}}{\beta _{1}}}} If β 2 ∼ Γ ( α 1 , β 1 ) {\displaystyle \beta _{2}\sim \Gamma (\alpha _{1},\beta _{1})} X 2 ∣ β 2 ∼ Γ ( α 2 , β 2 ) {\displaystyle X_{2}\mid \beta _{2}\sim \Gamma (\alpha _{2},\beta _{2})} X 2 ∼ β ′ ( α 2 , α 1 , 1 , β 1 ) {\displaystyle X_{2}\sim \beta '(\alpha _{2},\alpha _{1},1,\beta _{1})} β k {\displaystyle \beta _{k}} β 1 {\displaystyle \beta _{1}} β ′ ( p , 1 , a , b ) = Dagum ( p , a , b ) {\displaystyle \beta '(p,1,a,b)={\textrm {Dagum}}(p,a,b)} Dagum distribution β ′ ( 1 , p , a , b ) = SinghMaddala ( p , a , b ) {\displaystyle \beta '(1,p,a,b)={\textrm {SinghMaddala}}(p,a,b)} Singh–Maddala distribution . β ′ ( 1 , 1 , γ , σ ) = LL ( γ , σ ) {\displaystyle \beta '(1,1,\gamma ,\sigma )={\textrm {LL}}(\gamma ,\sigma )} 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}} α {\displaystyle \alpha } 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 } λ {\displaystyle \lambda } 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 } γ {\displaystyle \gamma } X 1 γ ∼ β ′ ( 1 , α ) {\displaystyle X^{\frac {1}{\gamma }}\sim \beta ^{\prime }(1,\alpha )} 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 )} ln X {\displaystyle \ln X} generalized logistic distribution . More generally, if X ∼ β ′ ( α , β , p , q ) {\displaystyle X\sim \beta '(\alpha ,\beta ,p,q)} ln X {\displaystyle \ln X} 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)} ± X {\displaystyle \pm {\sqrt {X}}} 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
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