In probability theory and statistics , the noncentral F -distribution is a continuous probability distribution that is a generalization of the (ordinary) F -distributionX /n 1 )/(Y /n 2 ), where the numerator X has a noncentral chi-squared distribution with n 1 degrees of freedom and the denominator Y has a central chi-squared distribution n 2 degrees of freedom. It is also required that X and Y are statistically independent of each other.
It is the distribution of the test statistic in analysis of variance problems when the null hypothesis is false. The noncentral F -distribution is used to find the power function of such a test.
Occurrence and specification
If
X
{\displaystyle X}
noncentral chi-squared random variable with noncentrality parameter
λ
{\displaystyle \lambda }
ν
1
{\displaystyle \nu _{1}}
Y
{\displaystyle Y}
chi-squared random variable with
ν
2
{\displaystyle \nu _{2}}
statistically independent of
X
{\displaystyle X}
F
=
X
/
ν
1
Y
/
ν
2
{\displaystyle F={\frac {X/\nu _{1}}{Y/\nu _{2}}}}
is a noncentral F -distributed random variable.
The probability density function (pdf) for the noncentral F -distribution is[ 1]
p
(
f
)
=
∑
k
=
0
∞
e
−
λ
/
2
(
λ
/
2
)
k
B
(
ν
2
2
,
ν
1
2
+
k
)
k
!
(
ν
1
ν
2
)
ν
1
2
+
k
(
ν
2
ν
2
+
ν
1
f
)
ν
1
+
ν
2
2
+
k
f
ν
1
/
2
−
1
+
k
{\displaystyle p(f)=\sum \limits _{k=0}^{\infty }{\frac {e^{-\lambda /2}(\lambda /2)^{k}}{B\left({\frac {\nu _{2}}{2}},{\frac {\nu _{1}}{2}}+k\right)k!}}\left({\frac {\nu _{1}}{\nu _{2}}}\right)^{{\frac {\nu _{1}}{2}}+k}\left({\frac {\nu _{2}}{\nu _{2}+\nu _{1}f}}\right)^{{\frac {\nu _{1}+\nu _{2}}{2}}+k}f^{\nu _{1}/2-1+k}}
when
f
≥
0
{\displaystyle f\geq 0}
ν
1
{\displaystyle \nu _{1}}
ν
2
{\displaystyle \nu _{2}}
λ
{\displaystyle \lambda }
B
(
x
,
y
)
{\displaystyle B(x,y)}
beta function , where
B
(
x
,
y
)
=
Γ
(
x
)
Γ
(
y
)
Γ
(
x
+
y
)
.
{\displaystyle B(x,y)={\frac {\Gamma (x)\Gamma (y)}{\Gamma (x+y)}}.}
The cumulative distribution function for the noncentral F -distribution is
F
(
x
|
d
1
,
d
2
,
λ
)
=
∑
j
=
0
∞
(
(
1
2
λ
)
j
j
!
e
−
λ
2
)
I
(
d
1
x
d
2
+
d
1
x
|
d
1
2
+
j
,
d
2
2
)
{\displaystyle F(x|d_{1},d_{2},\lambda )=\sum \limits _{j=0}^{\infty }\left({\frac {\left({\frac {1}{2}}\lambda \right)^{j}}{j!}}e^{-{\frac {\lambda }{2}}}\right)I\left({\frac {d_{1}x}{d_{2}+d_{1}x}}{\bigg |}{\frac {d_{1}}{2}}+j,{\frac {d_{2}}{2}}\right)}
where
I
{\displaystyle I}
regularized incomplete beta function .
The mean and variance of the noncentral F -distribution are
E
[
F
]
=
{
ν
2
(
ν
1
+
λ
)
ν
1
(
ν
2
−
2
)
ν
2
>
2
Does not exist
ν
2
≤
2
{\displaystyle \operatorname {E} \left[F\right]={\begin{cases}{\frac {\nu _{2}(\nu _{1}+\lambda )}{\nu _{1}(\nu _{2}-2)}}&\nu _{2}>2\\{\text{Does not exist}}&\nu _{2}\leq 2\\\end{cases}}}
and
Var
[
F
]
=
{
2
(
ν
1
+
λ
)
2
+
(
ν
1
+
2
λ
)
(
ν
2
−
2
)
(
ν
2
−
2
)
2
(
ν
2
−
4
)
(
ν
2
ν
1
)
2
ν
2
>
4
Does not exist
ν
2
≤
4.
{\displaystyle \operatorname {Var} \left[F\right]={\begin{cases}2{\frac {(\nu _{1}+\lambda )^{2}+(\nu _{1}+2\lambda )(\nu _{2}-2)}{(\nu _{2}-2)^{2}(\nu _{2}-4)}}\left({\frac {\nu _{2}}{\nu _{1}}}\right)^{2}&\nu _{2}>4\\{\text{Does not exist}}&\nu _{2}\leq 4.\\\end{cases}}}
Differential equation
This section
needs expansion . You can help by
adding to it .
(September 2014 )
The pdf of the noncentral F -distribution is a solution of the following differential equation :
{
4
x
(
ν
2
+
ν
1
x
)
2
f
″
(
x
)
+
f
′
(
x
)
(
−
2
ν
2
2
ν
1
+
8
ν
2
2
+
16
ν
1
2
x
2
+
4
ν
2
ν
1
2
x
2
−
2
λ
ν
2
ν
1
x
−
2
ν
2
ν
1
2
x
+
4
ν
2
2
ν
1
x
+
24
ν
2
ν
1
x
)
+
ν
1
(
ν
2
+
2
)
f
(
x
)
(
−
λ
ν
2
−
ν
2
ν
1
+
4
ν
2
+
4
ν
1
x
+
ν
2
ν
1
x
)
=
0
,
f
(
1
)
=
e
−
λ
/
2
ν
1
ν
1
2
ν
2
ν
2
2
(
ν
1
+
ν
2
)
1
2
(
−
ν
1
−
ν
2
)
1
F
1
(
1
2
(
ν
1
+
ν
2
)
;
ν
1
2
;
λ
ν
1
2
(
ν
1
+
ν
2
)
)
B
(
ν
1
2
,
ν
2
2
)
,
f
′
(
1
)
=
e
−
λ
/
2
ν
1
ν
1
2
ν
2
ν
2
2
(
ν
1
+
ν
2
)
1
2
(
−
ν
1
−
ν
2
−
2
)
(
ν
2
(
λ
1
F
1
(
1
2
(
ν
1
+
ν
2
+
2
)
;
1
2
(
ν
1
+
2
)
;
λ
ν
1
2
(
ν
1
+
ν
2
)
)
−
2
1
F
1
(
1
2
(
ν
1
+
ν
2
)
;
ν
1
2
;
λ
ν
1
2
(
ν
1
+
ν
2
)
)
)
−
2
ν
1
1
F
1
(
1
2
(
ν
1
+
ν
2
)
;
ν
1
2
;
λ
ν
1
2
(
ν
1
+
ν
2
)
)
)
2
B
(
ν
1
2
,
ν
2
2
)
}
{\displaystyle \left\{{\begin{array}{l}4x\left(\nu _{2}+\nu _{1}x\right){}^{2}f''(x)+f'(x)\left(-2\nu _{2}^{2}\nu _{1}+8\nu _{2}^{2}+16\nu _{1}^{2}x^{2}+4\nu _{2}\nu _{1}^{2}x^{2}-2\lambda \nu _{2}\nu _{1}x-2\nu _{2}\nu _{1}^{2}x+4\nu _{2}^{2}\nu _{1}x+24\nu _{2}\nu _{1}x\right)+\nu _{1}\left(\nu _{2}+2\right)f(x)\left(-\lambda \nu _{2}-\nu _{2}\nu _{1}+4\nu _{2}+4\nu _{1}x+\nu _{2}\nu _{1}x\right)=0,\\[12pt]f(1)={\frac {e^{-\lambda /2}\nu _{1}^{\frac {\nu _{1}}{2}}\nu _{2}^{\frac {\nu _{2}}{2}}\left(\nu _{1}+\nu _{2}\right){}^{{\frac {1}{2}}\left(-\nu _{1}-\nu _{2}\right)}\,_{1}F_{1}\left({\frac {1}{2}}\left(\nu _{1}+\nu _{2}\right);{\frac {\nu _{1}}{2}};{\frac {\lambda \nu _{1}}{2\left(\nu _{1}+\nu _{2}\right)}}\right)}{B\left({\frac {\nu _{1}}{2}},{\frac {\nu _{2}}{2}}\right)}},\\[12pt]f'(1)={\frac {e^{-\lambda /2}\nu _{1}^{\frac {\nu _{1}}{2}}\nu _{2}^{\frac {\nu _{2}}{2}}\left(\nu _{1}+\nu _{2}\right){}^{{\frac {1}{2}}\left(-\nu _{1}-\nu _{2}-2\right)}\left(\nu _{2}\left(\lambda \,_{1}F_{1}\left({\frac {1}{2}}\left(\nu _{1}+\nu _{2}+2\right);{\frac {1}{2}}\left(\nu _{1}+2\right);{\frac {\lambda \nu _{1}}{2\left(\nu _{1}+\nu _{2}\right)}}\right)-2\,_{1}F_{1}\left({\frac {1}{2}}\left(\nu _{1}+\nu _{2}\right);{\frac {\nu _{1}}{2}};{\frac {\lambda \nu _{1}}{2\left(\nu _{1}+\nu _{2}\right)}}\right)\right)-2\nu _{1}\,_{1}F_{1}\left({\frac {1}{2}}\left(\nu _{1}+\nu _{2}\right);{\frac {\nu _{1}}{2}};{\frac {\lambda \nu _{1}}{2\left(\nu _{1}+\nu _{2}\right)}}\right)\right)}{2B\left({\frac {\nu _{1}}{2}},{\frac {\nu _{2}}{2}}\right)}}\end{array}}\right\}}
Special cases
When λ = 0, the noncentral F -distribution becomes the
F -distribution
Z has a noncentral chi-squared distribution if
Z
=
lim
ν
2
→
∞
ν
1
F
{\displaystyle Z=\lim _{\nu _{2}\to \infty }\nu _{1}F}
where F has a noncentral F -distribution.
Implementations
The noncentral F -distribution is implemented in the R language (e.g., pf function), in MATLAB (ncfcdf, ncfinv, ncfpdf, ncfrnd and ncfstat functions in the statistics toolbox) in Mathematica (NoncentralFRatioDistribution function), in NumPy (random.noncentral_f), and in Boost C++ Libraries .[ 2]
A collaborative wiki page implements an interactive online calculator, programmed in the R language, for the noncentral t, chi-squared , and F distributions, at the Institute of Statistics and Econometrics, School of Business and Economics, Humboldt-Universität zu Berlin.[ 3]
Notes
References
External links
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
![{\displaystyle \operatorname {E} \left[F\right]={\begin{cases}{\frac {\nu _{2}(\nu _{1}+\lambda )}{\nu _{1}(\nu _{2}-2)}}&\nu _{2}>2\\{\text{Does not exist}}&\nu _{2}\leq 2\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41310d6c058293f139cfcff40c1be3f9993a7a1c)
![{\displaystyle \operatorname {Var} \left[F\right]={\begin{cases}2{\frac {(\nu _{1}+\lambda )^{2}+(\nu _{1}+2\lambda )(\nu _{2}-2)}{(\nu _{2}-2)^{2}(\nu _{2}-4)}}\left({\frac {\nu _{2}}{\nu _{1}}}\right)^{2}&\nu _{2}>4\\{\text{Does not exist}}&\nu _{2}\leq 4.\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a3d7bbf4e2f430838a76dfd13b72de4adaeab8)
![[icon]](/wiki/File:Wiki_letter_w_cropped.svg){kind=small}
![{\displaystyle \left\{{\begin{array}{l}4x\left(\nu _{2}+\nu _{1}x\right){}^{2}f''(x)+f'(x)\left(-2\nu _{2}^{2}\nu _{1}+8\nu _{2}^{2}+16\nu _{1}^{2}x^{2}+4\nu _{2}\nu _{1}^{2}x^{2}-2\lambda \nu _{2}\nu _{1}x-2\nu _{2}\nu _{1}^{2}x+4\nu _{2}^{2}\nu _{1}x+24\nu _{2}\nu _{1}x\right)+\nu _{1}\left(\nu _{2}+2\right)f(x)\left(-\lambda \nu _{2}-\nu _{2}\nu _{1}+4\nu _{2}+4\nu _{1}x+\nu _{2}\nu _{1}x\right)=0,\\[12pt]f(1)={\frac {e^{-\lambda /2}\nu _{1}^{\frac {\nu _{1}}{2}}\nu _{2}^{\frac {\nu _{2}}{2}}\left(\nu _{1}+\nu _{2}\right){}^{{\frac {1}{2}}\left(-\nu _{1}-\nu _{2}\right)}\,_{1}F_{1}\left({\frac {1}{2}}\left(\nu _{1}+\nu _{2}\right);{\frac {\nu _{1}}{2}};{\frac {\lambda \nu _{1}}{2\left(\nu _{1}+\nu _{2}\right)}}\right)}{B\left({\frac {\nu _{1}}{2}},{\frac {\nu _{2}}{2}}\right)}},\\[12pt]f'(1)={\frac {e^{-\lambda /2}\nu _{1}^{\frac {\nu _{1}}{2}}\nu _{2}^{\frac {\nu _{2}}{2}}\left(\nu _{1}+\nu _{2}\right){}^{{\frac {1}{2}}\left(-\nu _{1}-\nu _{2}-2\right)}\left(\nu _{2}\left(\lambda \,_{1}F_{1}\left({\frac {1}{2}}\left(\nu _{1}+\nu _{2}+2\right);{\frac {1}{2}}\left(\nu _{1}+2\right);{\frac {\lambda \nu _{1}}{2\left(\nu _{1}+\nu _{2}\right)}}\right)-2\,_{1}F_{1}\left({\frac {1}{2}}\left(\nu _{1}+\nu _{2}\right);{\frac {\nu _{1}}{2}};{\frac {\lambda \nu _{1}}{2\left(\nu _{1}+\nu _{2}\right)}}\right)\right)-2\nu _{1}\,_{1}F_{1}\left({\frac {1}{2}}\left(\nu _{1}+\nu _{2}\right);{\frac {\nu _{1}}{2}};{\frac {\lambda \nu _{1}}{2\left(\nu _{1}+\nu _{2}\right)}}\right)\right)}{2B\left({\frac {\nu _{1}}{2}},{\frac {\nu _{2}}{2}}\right)}}\end{array}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1149ba044e4c2413657b9d9fb59333cf30b923ff)
