- Source: Noncentral F-distribution
In probability theory and statistics, the noncentral F-distribution is a continuous probability distribution that is a noncentral generalization of the (ordinary) F-distribution. It describes the distribution of the quotient (X/n1)/(Y/n2), where the numerator X has a noncentral chi-squared distribution with n1 degrees of freedom and the denominator Y has a central chi-squared distribution with n2 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}
is a noncentral chi-squared random variable with noncentrality parameter
λ
{\displaystyle \lambda }
and
ν
1
{\displaystyle \nu _{1}}
degrees of freedom, and
Y
{\displaystyle Y}
is a chi-squared random variable with
ν
2
{\displaystyle \nu _{2}}
degrees of freedom that is statistically independent of
X
{\displaystyle X}
, then
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
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}
and zero otherwise.
The degrees of freedom
ν
1
{\displaystyle \nu _{1}}
and
ν
2
{\displaystyle \nu _{2}}
are positive.
The term
B
(
x
,
y
)
{\displaystyle B(x,y)}
is the 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\mid d_{1},d_{2},\lambda )=\sum \limits _{j=0}^{\infty }\left({\frac {\left({\frac {1}{2}}\lambda \right)^{j}}{j!}}e^{-\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}
is the 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
)
if
ν
2
>
2
does not exist
if
ν
2
≤
2
{\displaystyle \operatorname {E} [F]\quad {\begin{cases}={\frac {\nu _{2}(\nu _{1}+\lambda )}{\nu _{1}(\nu _{2}-2)}}&{\text{if }}\nu _{2}>2\\{\text{does not exist}}&{\text{if }}\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
if
ν
2
>
4
does not exist
if
ν
2
≤
4.
{\displaystyle \operatorname {Var} [F]\quad {\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}&{\text{if }}\nu _{2}>4\\{\text{does not exist}}&{\text{if }}\nu _{2}\leq 4.\\\end{cases}}}
Special cases
When λ = 0, the noncentral F-distribution becomes the
F-distribution.
Related distributions
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.
See also noncentral t-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.
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 of the Humboldt University of Berlin.
Notes
References
Weisstein, Eric W.; et al. "Noncentral F-distribution". MathWorld. Wolfram Research, Inc. Retrieved 20 August 2011.
Kata Kunci Pencarian:
- Noncentral F-distribution
- F-distribution
- Noncentral beta distribution
- Noncentral t-distribution
- Noncentral chi-squared distribution
- Noncentral distribution
- Noncentral chi distribution
- Fisher's noncentral hypergeometric distribution
- List of probability distributions
- Wallenius' noncentral hypergeometric distribution