Noncentral F-distribution

If $X$ is a noncentral chi-squared random variable with noncentrality parameter $\backslash lambda$ and $\backslash nu\_1$ degrees of freedom, and $Y$ is a chi-squared random variable with $\backslash nu\_2$ degrees of freedom that is statistically independent of $X$, then

:$$

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{{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 }}

:$$

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}

when $f\backslash ge0$ and zero otherwise.

The degrees of freedom $\backslash nu\_1$ and $\backslash nu\_2$ are positive.

The term $B(x,y)$ is the beta function, where

:$$

B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}.

The cumulative distribution function for the noncentral F-distribution is

:$$

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)

where $I$ is the regularized incomplete beta function.

The mean and variance of the noncentral F-distribution are

:$$

\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}

and

:$$

\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}

When λ = 0, the noncentral F-distribution becomes the

F-distribution.

Z has a noncentral chi-squared distribution if

: $Z=\backslash lim\_\{\backslash nu\_2\backslash to\backslash infty\}\backslash nu\_1\; F$

where F has a noncentral F-distribution.

See also noncentral t-distribution.

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.{{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}}

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.{{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}}

• {{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

F