Noncentral Beta Notation
Beta(α あるふぁ , β べーた , λ らむだ ) Parameters
α あるふぁ > 0 shape (real )β べーた > 0 shape (real )λ らむだ ≥ 0 noncentrality (real ) Support
x
∈
[
0
;
1
]
{\displaystyle x\in [0;1]\!}
PDF
(type I)
∑
j
=
0
∞
e
−
λ らむだ
/
2
(
λ らむだ
2
)
j
j
!
x
α あるふぁ
+
j
−
1
(
1
−
x
)
β べーた
−
1
B
(
α あるふぁ
+
j
,
β べーた
)
{\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 らむだ
/
2
(
λ らむだ
2
)
j
j
!
I
x
(
α あるふぁ
+
j
,
β べーた
)
{\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 らむだ
2
Γ がんま
(
α あるふぁ
+
1
)
Γ がんま
(
α あるふぁ
)
Γ がんま
(
α あるふぁ
+
β べーた
)
Γ がんま
(
α あるふぁ
+
β べーた
+
1
)
2
F
2
(
α あるふぁ
+
β べーた
,
α あるふぁ
+
1
;
α あるふぁ
,
α あるふぁ
+
β べーた
+
1
;
λ らむだ
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 らむだ
2
Γ がんま
(
α あるふぁ
+
2
)
Γ がんま
(
α あるふぁ
)
Γ がんま
(
α あるふぁ
+
β べーた
)
Γ がんま
(
α あるふぁ
+
β べーた
+
2
)
2
F
2
(
α あるふぁ
+
β べーた
,
α あるふぁ
+
2
;
α あるふぁ
,
α あるふぁ
+
β べーた
+
2
;
λ らむだ
2
)
−
μ みゅー
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 みゅー
{\displaystyle \mu }
is the mean. (see 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 かい
m
2
(
λ らむだ
)
χ かい
m
2
(
λ らむだ
)
+
χ かい
n
2
,
{\displaystyle X={\frac {\chi _{m}^{2}(\lambda )}{\chi _{m}^{2}(\lambda )+\chi _{n}^{2}}},}
where
χ かい
m
2
(
λ らむだ
)
{\displaystyle \chi _{m}^{2}(\lambda )}
is a
noncentral chi-squared random variable with degrees of freedom m and noncentrality parameter
λ らむだ
{\displaystyle \lambda }
, and
χ かい
n
2
{\displaystyle \chi _{n}^{2}}
is a central chi-squared random variable with degrees of freedom n , independent of
χ かい
m
2
(
λ らむだ
)
{\displaystyle \chi _{m}^{2}(\lambda )}
.[1]
In this case,
X
∼
Beta
(
m
2
,
n
2
,
λ らむだ
)
{\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 かい
n
2
χ かい
n
2
+
χ かい
m
2
(
λ らむだ
)
,
{\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 [ edit ]
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 あるふぁ
+
j
,
β べーた
)
,
{\displaystyle F(x)=\sum _{j=0}^{\infty }P(j)I_{x}(\alpha +j,\beta ),}
where λ らむだ is the noncentrality parameter, P (.) is the Poisson(λ らむだ /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 らむだ
2
)
j
e
−
λ らむだ
/
2
I
x
(
α あるふぁ
+
j
,
β べーた
)
.
{\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 あるふぁ
,
β べーた
+
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 [ edit ]
The (Type I) probability density function for the noncentral beta distribution is:
f
(
x
)
=
∑
j
=
0
∞
1
j
!
(
λ らむだ
2
)
j
e
−
λ らむだ
/
2
x
α あるふぁ
+
j
−
1
(
1
−
x
)
β べーた
−
1
B
(
α あるふぁ
+
j
,
β べーた
)
.
{\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 あるふぁ
{\displaystyle \alpha }
and
β べーた
{\displaystyle \beta }
are the shape parameters, and
λ らむだ
{\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 [ edit ]
Transformations [ edit ]
If
X
∼
Beta
(
α あるふぁ
,
β べーた
,
λ らむだ
)
{\displaystyle X\sim {\mbox{Beta}}\left(\alpha ,\beta ,\lambda \right)}
, then
β べーた
X
α あるふぁ
(
1
−
X
)
{\displaystyle {\frac {\beta X}{\alpha (1-X)}}}
follows a noncentral F-distribution with
2
α あるふぁ
,
2
β べーた
{\displaystyle 2\alpha ,2\beta }
degrees of freedom, and non-centrality parameter
λ らむだ
{\displaystyle \lambda }
.
If
X
{\displaystyle X}
follows a noncentral F-distribution
F
μ みゅー
1
,
μ みゅー
2
(
λ らむだ
)
{\displaystyle F_{\mu _{1},\mu _{2}}\left(\lambda \right)}
with
μ みゅー
1
{\displaystyle \mu _{1}}
numerator degrees of freedom and
μ みゅー
2
{\displaystyle \mu _{2}}
denominator degrees of freedom, then
Z
=
μ みゅー
2
μ みゅー
1
μ みゅー
2
μ みゅー
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 みゅー
1
,
1
2
μ みゅー
2
,
λ らむだ
)
{\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 [ edit ]
When
λ らむだ
=
0
{\displaystyle \lambda =0}
, the noncentral beta distribution is equivalent to the (central) beta distribution .
References [ edit ]
Citations [ edit ]
Sources [ edit ]
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