Noncentral Beta Notation
Beta(α, β, λ) Parameters
α > 0 Support
x
∈
[
0
;
1
]
{\displaystyle x\in [0;1]\!}
PDF
(type I)
∑
j
=
0
∞
e
−
λ
/
2
(
λ
2
)
j
j
!
x
α
+
j
−
1
(
1
−
x
)
β
−
1
B
(
α
+
j
,
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
/
2
(
λ
2
)
j
j
!
I
x
(
α
+
j
,
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
2
Γ
(
α
+
1
)
Γ
(
α
)
Γ
(
α
+
β
)
Γ
(
α
+
β
+
1
)
2
F
2
(
α
+
β
,
α
+
1
;
α
,
α
+
β
+
1
;
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)}
(see Confluent hypergeometric function ) Variance
(type I)
e
−
λ
2
Γ
(
α
+
2
)
Γ
(
α
)
Γ
(
α
+
β
)
Γ
(
α
+
β
+
2
)
2
F
2
(
α
+
β
,
α
+
2
;
α
,
α
+
β
+
2
;
λ
2
)
−
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}}
where
μ
{\displaystyle \mu }
is the mean. (see Confluent hypergeometric function )
In
.
The noncentral beta distribution (Type I) is the distribution of the ratio
X
=
χ
m
2
(
λ
)
χ
m
2
(
λ
)
+
χ
n
2
,
{\displaystyle X={\frac {\chi _{m}^{2}(\lambda )}{\chi _{m}^{2}(\lambda )+\chi _{n}^{2}}},}
where
χ
m
2
(
λ
)
{\displaystyle \chi _{m}^{2}(\lambda )}
is a
noncentral chi-squared random variable with degrees of freedom m and noncentrality parameter
λ
{\displaystyle \lambda }
, and
χ
n
2
{\displaystyle \chi _{n}^{2}}
is a central chi-squared random variable with degrees of freedom n , independent of
χ
m
2
(
λ
)
{\displaystyle \chi _{m}^{2}(\lambda )}
.[1]
In this case,
X
∼
Beta
(
m
2
,
n
2
,
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
n
2
χ
n
2
+
χ
m
2
(
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] If
Y
{\displaystyle Y}
follows
the type II distribution, then
X
=
1
−
Y
{\displaystyle X=1-Y}
follows a type I distribution.
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
+
j
,
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)}
is the
incomplete beta function
. That is,
F
(
x
)
=
∑
j
=
0
∞
1
j
!
(
λ
2
)
j
e
−
λ
/
2
I
x
(
α
+
j
,
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
,
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] and Chattamvelli.[1]
Probability density function
The (Type I) probability density function for the noncentral beta distribution is:
f
(
x
)
=
∑
j
=
0
∞
1
j
!
(
λ
2
)
j
e
−
λ
/
2
x
α
+
j
−
1
(
1
−
x
)
β
−
1
B
(
α
+
j
,
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}
is the beta function ,
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
are the shape parameters, and
λ
{\displaystyle \lambda }
is the
noncentrality parameter. The density of
Y is the same as that of
1-X with the degrees of freedom reversed.
[1]
Related distributions
Transformations
If
X
∼
Beta
(
α
,
β
,
λ
)
{\displaystyle X\sim {\mbox{Beta}}\left(\alpha ,\beta ,\lambda \right)}
, then
β
X
α
(
1
−
X
)
{\displaystyle {\frac {\beta X}{\alpha (1-X)}}}
follows a noncentral F-distribution with
2
α
,
2
β
{\displaystyle 2\alpha ,2\beta }
degrees of freedom, and non-centrality parameter
λ
{\displaystyle \lambda }
.
If
X
{\displaystyle X}
follows a noncentral F-distribution
F
μ
1
,
μ
2
(
λ
)
{\displaystyle F_{\mu _{1},\mu _{2}}\left(\lambda \right)}
with
μ
1
{\displaystyle \mu _{1}}
numerator degrees of freedom and
μ
2
{\displaystyle \mu _{2}}
denominator degrees of freedom, then
Z
=
μ
2
μ
1
μ
2
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
1
,
1
2
μ
2
,
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
=
0
{\displaystyle \lambda =0}
, the noncentral beta distribution is equivalent to the (central) beta distribution .
References
Citations
Sources
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