Noncentral Beta Notation
Beta(α, β, λ) Parameters
α > 0 shape (real )shape (real )noncentrality (real ) Support
x
∈ ∈ -->
[
0
;
1
]
{\displaystyle x\in [0;1]\!}
PDF
(type I)
∑ ∑ -->
j
=
0
∞ ∞ -->
e
− − -->
λ λ -->
/
2
(
λ λ -->
2
)
j
j
!
x
α α -->
+
j
− − -->
1
(
1
− − -->
x
)
β β -->
− − -->
1
B
(
α α -->
+
j
,
β β -->
)
{\displaystyle \sum _{j=0}^{\infty }e^{-\lambda /2}{\frac {\left({\frac {\lambda }{2}}\right)^{j}}{j!}}{\frac {x^{\alpha +j-1}\left(1-x\right)^{\beta -1}}{\mathrm {B} \left(\alpha +j,\beta \right)}}}
CDF
(type I)
∑ ∑ -->
j
=
0
∞ ∞ -->
e
− − -->
λ λ -->
/
2
(
λ λ -->
2
)
j
j
!
I
x
(
α α -->
+
j
,
β β -->
)
{\displaystyle \sum _{j=0}^{\infty }e^{-\lambda /2}{\frac {\left({\frac {\lambda }{2}}\right)^{j}}{j!}}I_{x}\left(\alpha +j,\beta \right)}
Mean
(type I)
e
− − -->
λ λ -->
2
Γ Γ -->
(
α α -->
+
1
)
Γ Γ -->
(
α α -->
)
Γ Γ -->
(
α α -->
+
β β -->
)
Γ Γ -->
(
α α -->
+
β β -->
+
1
)
2
F
2
(
α α -->
+
β β -->
,
α α -->
+
1
;
α α -->
,
α α -->
+
β β -->
+
1
;
λ λ -->
2
)
{\displaystyle e^{-{\frac {\lambda }{2}}}{\frac {\Gamma \left(\alpha +1\right)}{\Gamma \left(\alpha \right)}}{\frac {\Gamma \left(\alpha +\beta \right)}{\Gamma \left(\alpha +\beta +1\right)}}{}_{2}F_{2}\left(\alpha +\beta ,\alpha +1;\alpha ,\alpha +\beta +1;{\frac {\lambda }{2}}\right)}
Confluent hypergeometric function ) Variance
(type I)
e
− − -->
λ λ -->
2
Γ Γ -->
(
α α -->
+
2
)
Γ Γ -->
(
α α -->
)
Γ Γ -->
(
α α -->
+
β β -->
)
Γ Γ -->
(
α α -->
+
β β -->
+
2
)
2
F
2
(
α α -->
+
β β -->
,
α α -->
+
2
;
α α -->
,
α α -->
+
β β -->
+
2
;
λ λ -->
2
)
− − -->
μ μ -->
2
{\displaystyle e^{-{\frac {\lambda }{2}}}{\frac {\Gamma \left(\alpha +2\right)}{\Gamma \left(\alpha \right)}}{\frac {\Gamma \left(\alpha +\beta \right)}{\Gamma \left(\alpha +\beta +2\right)}}{}_{2}F_{2}\left(\alpha +\beta ,\alpha +2;\alpha ,\alpha +\beta +2;{\frac {\lambda }{2}}\right)-\mu ^{2}}
μ μ -->
{\displaystyle \mu }
Confluent hypergeometric function )
In probability theory and statistics , the noncentral beta distribution is a continuous probability distribution that is a noncentral generalization of the (central) beta distribution .
The noncentral beta distribution (Type I) is the distribution of the ratio
X
=
χ χ -->
m
2
(
λ λ -->
)
χ χ -->
m
2
(
λ λ -->
)
+
χ χ -->
n
2
,
{\displaystyle X={\frac {\chi _{m}^{2}(\lambda )}{\chi _{m}^{2}(\lambda )+\chi _{n}^{2}}},}
where
χ χ -->
m
2
(
λ λ -->
)
{\displaystyle \chi _{m}^{2}(\lambda )}
noncentral chi-squared random variable with degrees of freedom m and noncentrality parameter
λ λ -->
{\displaystyle \lambda }
χ χ -->
n
2
{\displaystyle \chi _{n}^{2}}
chi-squared random variable with degrees of freedom n , independent of
χ χ -->
m
2
(
λ λ -->
)
{\displaystyle \chi _{m}^{2}(\lambda )}
[ 1]
X
∼ ∼ -->
Beta
(
m
2
,
n
2
,
λ λ -->
)
{\displaystyle X\sim {\mbox{Beta}}\left({\frac {m}{2}},{\frac {n}{2}},\lambda \right)}
A Type II noncentral beta distribution is the distribution
of the ratio
Y
=
χ χ -->
n
2
χ χ -->
n
2
+
χ χ -->
m
2
(
λ λ -->
)
,
{\displaystyle Y={\frac {\chi _{n}^{2}}{\chi _{n}^{2}+\chi _{m}^{2}(\lambda )}},}
where the noncentral chi-squared variable is in the denominator only.[ 1]
Y
{\displaystyle Y}
X
=
1
− − -->
Y
{\displaystyle X=1-Y}
Cumulative distribution function
The Type I cumulative distribution function is usually represented as a Poisson mixture of central beta random variables:[ 1]
F
(
x
)
=
∑ ∑ -->
j
=
0
∞ ∞ -->
P
(
j
)
I
x
(
α α -->
+
j
,
β β -->
)
,
{\displaystyle F(x)=\sum _{j=0}^{\infty }P(j)I_{x}(\alpha +j,\beta ),}
where λ is the noncentrality parameter, P (.) is the Poisson(λ/2) probability mass function, \alpha=m/2 and \beta=n/2 are shape parameters, and
I
x
(
a
,
b
)
{\displaystyle I_{x}(a,b)}
incomplete beta function . That is,
F
(
x
)
=
∑ ∑ -->
j
=
0
∞ ∞ -->
1
j
!
(
λ λ -->
2
)
j
e
− − -->
λ λ -->
/
2
I
x
(
α α -->
+
j
,
β β -->
)
.
{\displaystyle F(x)=\sum _{j=0}^{\infty }{\frac {1}{j!}}\left({\frac {\lambda }{2}}\right)^{j}e^{-\lambda /2}I_{x}(\alpha +j,\beta ).}
The Type II cumulative distribution function in mixture form is
F
(
x
)
=
∑ ∑ -->
j
=
0
∞ ∞ -->
P
(
j
)
I
x
(
α α -->
,
β β -->
+
j
)
.
{\displaystyle F(x)=\sum _{j=0}^{\infty }P(j)I_{x}(\alpha ,\beta +j).}
Algorithms for evaluating the noncentral beta distribution functions are given by Posten[ 2] [ 1]
Probability density function
The (Type I) probability density function for the noncentral beta distribution is:
f
(
x
)
=
∑ ∑ -->
j
=
0
∞ ∞ -->
1
j
!
(
λ λ -->
2
)
j
e
− − -->
λ λ -->
/
2
x
α α -->
+
j
− − -->
1
(
1
− − -->
x
)
β β -->
− − -->
1
B
(
α α -->
+
j
,
β β -->
)
.
{\displaystyle f(x)=\sum _{j=0}^{\infty }{\frac {1}{j!}}\left({\frac {\lambda }{2}}\right)^{j}e^{-\lambda /2}{\frac {x^{\alpha +j-1}(1-x)^{\beta -1}}{B(\alpha +j,\beta )}}.}
where
B
{\displaystyle B}
beta function ,
α α -->
{\displaystyle \alpha }
β β -->
{\displaystyle \beta }
λ λ -->
{\displaystyle \lambda }
noncentrality parameter . The density of Y is the same as that of 1-X with the degrees of freedom reversed.[ 1]
If
X
∼ ∼ -->
Beta
(
α α -->
,
β β -->
,
λ λ -->
)
{\displaystyle X\sim {\mbox{Beta}}\left(\alpha ,\beta ,\lambda \right)}
β β -->
X
α α -->
(
1
− − -->
X
)
{\displaystyle {\frac {\beta X}{\alpha (1-X)}}}
noncentral F-distribution with
2
α α -->
,
2
β β -->
{\displaystyle 2\alpha ,2\beta }
λ λ -->
{\displaystyle \lambda }
If
X
{\displaystyle X}
noncentral F-distribution
F
μ μ -->
1
,
μ μ -->
2
(
λ λ -->
)
{\displaystyle F_{\mu _{1},\mu _{2}}\left(\lambda \right)}
μ μ -->
1
{\displaystyle \mu _{1}}
μ μ -->
2
{\displaystyle \mu _{2}}
Z
=
μ μ -->
2
μ μ -->
1
μ μ -->
2
μ μ -->
1
+
X
− − -->
1
{\displaystyle Z={\cfrac {\cfrac {\mu _{2}}{\mu _{1}}}{{\cfrac {\mu _{2}}{\mu _{1}}}+X^{-1}}}}
follows a noncentral Beta distribution:
Z
∼ ∼ -->
Beta
(
1
2
μ μ -->
1
,
1
2
μ μ -->
2
,
λ λ -->
)
{\displaystyle Z\sim {\mbox{Beta}}\left({\frac {1}{2}}\mu _{1},{\frac {1}{2}}\mu _{2},\lambda \right)}
This is derived from making a straightforward transformation.
Special cases
When
λ λ -->
=
0
{\displaystyle \lambda =0}
beta distribution .
References
Citations
Sources
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
![{\displaystyle x\in [0;1]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/394f69db847ba283727b0bc73bccc019572a72ae)
