Editing Noncentral F-distribution

{{Short description|Probability distribution generalizing the F-distribution with a noncentrality parameter}}
{{DISPLAYTITLE:Noncentral ''F''-distribution}}
In [[probability theory]] and [[statistics]], the '''noncentral ''F''-distribution''' is a [[continuous  probability distribution]] that is a [[noncentral distribution|noncentral generalization]] of the (ordinary) [[F-distribution|''F''-distribution]]. It describes the distribution of the quotient (''X''/''n''<sub>1</sub>)/(''Y''/''n''<sub>2</sub>), where the numerator ''X'' has a [[noncentral chi-squared distribution]] with ''n''<sub>1</sub> degrees of freedom and the denominator ''Y'' has a central [[chi-squared distribution]] with ''n''<sub>2</sub> degrees of freedom. It is also required that ''X'' and ''Y'' are [[statistical independence|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 [[statistical power|power function]] of such a test.

== Occurrence and specification ==
If <math>X</math> is a [[Noncentral chi-squared distribution|noncentral chi-squared]] random variable with noncentrality parameter <math>\lambda</math> and <math>\nu_1</math> degrees of freedom, and <math>Y</math> is a [[Chi-squared distribution|chi-squared]] random variable with <math>\nu_2</math> degrees of freedom that is [[statistical independence|statistically independent]] of <math>X</math>, then
:<math>
F=\frac{X/\nu_1}{Y/\nu_2}
</math>
is a noncentral ''F''-distributed random variable.
The [[probability density function]] (pdf) for the noncentral ''F''-distribution is<ref>{{cite book |first=S. |last=Kay |title=Fundamentals of Statistical Signal Processing: Detection Theory |location=New Jersey |publisher=Prentice Hall |year=1998 |page=29 |isbn=0-13-504135-X }}</ref>
:<math>
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_1f}\right)^{\frac{\nu_1+\nu_2}{2}+k}f^{\nu_1/2-1+k}
 </math>
when <math>f\ge0</math> and zero otherwise.
The degrees of freedom <math>\nu_1</math> and <math>\nu_2</math> are positive.
The term <math>B(x,y)</math> is the [[beta function]], where
:<math>
B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}.
</math>

The [[cumulative distribution function]] for the noncentral ''F''-distribution is
:<math>
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_1x}{d_2 + d_1x}\bigg|\frac{d_1}{2}+j,\frac{d_2}{2}\right)
</math>
where <math>I</math> is the [[regularized incomplete beta function]].

The mean and variance of the noncentral ''F''-distribution are
:<math>
\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\le2\\
\end{cases}
</math>
and
:<math>
\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\le4.\\
\end{cases}
</math>

== Special cases ==
When ''λ''&nbsp;=&nbsp;0, the noncentral ''F''-distribution becomes the
[[F-distribution|''F''-distribution]].

== Related distributions ==
''Z'' has a [[noncentral chi-squared distribution]] if

: <math> Z=\lim_{\nu_2\to\infty}\nu_1 F </math>

where ''F'' has a noncentral ''F''-distribution.

See also [[noncentral t-distribution]].

== Implementations ==
The noncentral ''F''-distribution is implemented in the [[R (programming language)|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]].<ref>{{cite web |url=http://www.boost.org/doc/libs/1_39_0/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_f_dist.html |title=Noncentral F Distribution: Boost 1.39.0 |author=John Maddock |author2=Paul A. Bristow |author3=Hubert Holin |author4=Xiaogang Zhang |author5=Bruno Lalande |author6=Johan Råde |work=Boost.org |access-date=20 August 2011}}</ref>

A collaborative wiki page implements an interactive online calculator, programmed in the [[R (programming language)|R language]], for the noncentral t, [[Noncentral chi-squared distribution|chi-squared]], and F distributions, at the Institute of Statistics and Econometrics of the [[Humboldt University of Berlin]].<ref>{{cite web |url=http://mars.wiwi.hu-berlin.de/mediawiki/slides/index.php/Comparison_of_noncentral_and_central_distributions |title=Comparison of noncentral and central distributions |author=Sigbert Klinke |date=10 December 2008 |publisher=Humboldt-Universität zu Berlin}}</ref>

== Notes ==
<references/>

== References ==
* {{cite web |url=http://mathworld.wolfram.com/NoncentralF-Distribution.html |title=Noncentral ''F''-distribution |first=Eric W.|last=Weisstein |author-link=Eric W. Weisstein |work=[[MathWorld]] |publisher=Wolfram Research, Inc |access-date=20 August 2011|display-authors=etal}}

{{Probability distributions}}

[[Category:Continuous distributions]]
[[Category:Noncentral distributions|F]]