- Source: Noncentral beta distribution
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
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 )}
.
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. 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:
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 and Chattamvelli.
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.
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
=Kata Kunci Pencarian:
- Noncentral beta distribution
- Noncentral F-distribution
- Noncentral distribution
- Beta distribution
- Noncentral t-distribution
- Chi-squared distribution
- F-distribution
- List of probability distributions
- Distribution of the product of two random variables
- Student's t-distribution
