Probability distribution
Probability density function
Cumulative distribution function
Parameters
α
>
0
{\displaystyle \alpha >0}
shape (real )
β
>
0
{\displaystyle \beta >0}
shape (real) 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 }}{B(\alpha ,\beta )}}\!}
CDF
I
x
1
+
x
(
α
,
β
)
{\displaystyle I_{{\frac {x}{1+x}}(\alpha ,\beta )}}
where
I
x
(
α
,
β
)
{\displaystyle I_{x}(\alpha ,\beta )}
is the incomplete beta function Mean
α
β
−
1
if
β
>
1
{\displaystyle {\frac {\alpha }{\beta -1}}{\text{ if }}\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
if
β
>
2
{\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}{\text{ if }}\beta >2}
Skewness
2
(
2
α
+
β
−
1
)
β
−
3
β
−
2
α
(
α
+
β
−
1
)
if
β
>
3
{\displaystyle {\frac {2(2\alpha +\beta -1)}{\beta -3}}{\sqrt {\frac {\beta -2}{\alpha (\alpha +\beta -1)}}}{\text{ if }}\beta >3}
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] ) is an absolutely continuous probability distribution . If
p
∈
[
0
,
1
]
{\displaystyle p\in [0,1]}
has a beta distribution , then the odds
p
1
−
p
{\displaystyle {\frac {p}{1-p}}}
has a beta prime distribution.
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 }}{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 .
The expected value, variance, and other details of the distribution are given in the sidebox; for
β
>
4
{\displaystyle \beta >4}
, the excess kurtosis is
γ
2
=
6
α
(
α
+
β
−
1
)
(
5
β
−
11
)
+
(
β
−
1
)
2
(
β
−
2
)
α
(
α
+
β
−
1
)
(
β
−
3
)
(
β
−
4
)
.
{\displaystyle \gamma _{2}=6{\frac {\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
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 {B(\alpha +k,\beta -k)}{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 B(\alpha ,\beta )}}}
where
2
F
1
{\displaystyle {}_{2}F_{1}}
is the Gauss's hypergeometric function 2 F1 .
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 ) 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 }}{qB(\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 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 :
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 q 2 .
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 )}
If
X
1
∼
β
′
(
α
,
β
)
{\displaystyle X_{1}\sim \beta '(\alpha ,\beta )}
and
X
2
∼
β
′
(
α
,
β
)
{\displaystyle X_{2}\sim \beta '(\alpha ,\beta )}
two iid variables, then
Y
=
X
1
+
X
2
∼
β
′
(
γ
,
δ
)
{\displaystyle Y=X_{1}+X_{2}\sim \beta '(\gamma ,\delta )}
with
γ
=
2
α
(
α
+
β
2
−
2
β
+
2
α
β
−
4
α
+
1
)
(
β
−
1
)
(
α
+
β
−
1
)
{\displaystyle \gamma ={\frac {2\alpha (\alpha +\beta ^{2}-2\beta +2\alpha \beta -4\alpha +1)}{(\beta -1)(\alpha +\beta -1)}}}
and
δ
=
2
α
+
β
2
−
β
+
2
α
β
−
4
α
α
+
β
−
1
{\displaystyle \delta ={\frac {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
{\displaystyle X_{1},...,X_{n}n}
iid variables following the same beta prime distribution, i.e.
∀
i
,
1
≤
i
≤
n
,
X
i
∼
β
′
(
α
,
β
)
{\displaystyle \forall i,1\leq i\leq n,X_{i}\sim \beta '(\alpha ,\beta )}
, then the sum
S
=
X
1
+
.
.
.
+
X
n
∼
β
′
(
γ
,
δ
)
{\displaystyle S=X_{1}+...+X_{n}\sim \beta '(\gamma ,\delta )}
with
γ
=
n
α
(
α
+
β
2
−
2
β
+
n
α
β
−
2
n
α
+
1
)
(
β
−
1
)
(
α
+
β
−
1
)
{\displaystyle \gamma ={\frac {n\alpha (\alpha +\beta ^{2}-2\beta +n\alpha \beta -2n\alpha +1)}{(\beta -1)(\alpha +\beta -1)}}}
and
δ
=
2
α
+
β
2
−
β
+
n
α
β
−
2
n
α
α
+
β
−
1
{\displaystyle \delta ={\frac {2\alpha +\beta ^{2}-\beta +n\alpha \beta -2n\alpha }{\alpha +\beta -1}}}
.
Related distributions
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 }})}
.
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 )}
.
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 )}
.
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
α
1
,
α
2
,
θ
1
θ
2
{\displaystyle \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
∼
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.
Notes
References
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
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint) Directional Degenerate and singular Families
This page was last edited on 27 April 2024, at 21:08