# Noncentral F-distribution

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In probability theory and statistics, the noncentral F-distribution is a continuous probability distribution that is a 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 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 ${\displaystyle X}$ is a noncentral chi-squared random variable with noncentrality parameter ${\displaystyle \lambda }$ and ${\displaystyle \nu _{1}}$ degrees of freedom, and ${\displaystyle Y}$ is a chi-squared random variable with ${\displaystyle \nu _{2}}$ degrees of freedom that is statistically independent of ${\displaystyle X}$, then

${\displaystyle F={\frac {X/\nu _{1}}{Y/\nu _{2}}}}$

is a noncentral F-distributed random variable. The probability density function (pdf) for the noncentral F-distribution is[1]

${\displaystyle p(f)=\sum \limits _{k=0}^{\infty }{\frac {e^{-\lambda /2}(\lambda /2)^{k}}{B\left({\frac {\nu _{2}}{2}},{\frac {\nu _{1}}{2}}+k\right)k!}}\left({\frac {\nu _{1}}{\nu _{2}}}\right)^{{\frac {\nu _{1}}{2}}+k}\left({\frac {\nu _{2}}{\nu _{2}+\nu _{1}f}}\right)^{{\frac {\nu _{1}+\nu _{2}}{2}}+k}f^{\nu _{1}/2-1+k}}$

when ${\displaystyle f\geq 0}$ and zero otherwise. The degrees of freedom ${\displaystyle \nu _{1}}$ and ${\displaystyle \nu _{2}}$ are positive. The noncentrality parameter ${\displaystyle \lambda }$ is nonnegative. The term ${\displaystyle B(x,y)}$ is the beta function, where

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

The cumulative distribution function for the noncentral F-distribution is

${\displaystyle F(x|d_{1},d_{2},\lambda )=\sum \limits _{j=0}^{\infty }\left({\frac {\left({\frac {1}{2}}\lambda \right)^{j}}{j!}}e^{-{\frac {\lambda }{2}}}\right)I\left({\frac {d_{1}x}{d_{2}+d_{1}x}}{\bigg |}{\frac {d_{1}}{2}}+j,{\frac {d_{2}}{2}}\right)}$

where ${\displaystyle I}$ is the regularized incomplete beta function.

The mean and variance of the noncentral F-distribution are

${\displaystyle \operatorname {E} \left[F\right]={\begin{cases}{\frac {\nu _{2}(\nu _{1}+\lambda )}{\nu _{1}(\nu _{2}-2)}}&\nu _{2}>2\\{\text{Does not exist}}&\nu _{2}\leq 2\\\end{cases}}}$

and

${\displaystyle \operatorname {Var} \left[F\right]={\begin{cases}2{\frac {(\nu _{1}+\lambda )^{2}+(\nu _{1}+2\lambda )(\nu _{2}-2)}{(\nu _{2}-2)^{2}(\nu _{2}-4)}}\left({\frac {\nu _{2}}{\nu _{1}}}\right)^{2}&\nu _{2}>4\\{\text{Does not exist}}&\nu _{2}\leq 4.\\\end{cases}}}$

### Differential equation

The pdf of the noncentral F-distribution is a solution of the following differential equation:

${\displaystyle \left\{{\begin{array}{l}4x\left(\nu _{2}+\nu _{1}x\right){}^{2}f''(x)+f'(x)\left(-2\nu _{2}^{2}\nu _{1}+8\nu _{2}^{2}+16\nu _{1}^{2}x^{2}+4\nu _{2}\nu _{1}^{2}x^{2}-2\lambda \nu _{2}\nu _{1}x-2\nu _{2}\nu _{1}^{2}x+4\nu _{2}^{2}\nu _{1}x+24\nu _{2}\nu _{1}x\right)+\nu _{1}\left(\nu _{2}+2\right)f(x)\left(-\lambda \nu _{2}-\nu _{2}\nu _{1}+4\nu _{2}+4\nu _{1}x+\nu _{2}\nu _{1}x\right)=0,\\[12pt]f(1)={\frac {e^{-\lambda /2}\nu _{1}^{\frac {\nu _{1}}{2}}\nu _{2}^{\frac {\nu _{2}}{2}}\left(\nu _{1}+\nu _{2}\right){}^{{\frac {1}{2}}\left(-\nu _{1}-\nu _{2}\right)}\,_{1}F_{1}\left({\frac {1}{2}}\left(\nu _{1}+\nu _{2}\right);{\frac {\nu _{1}}{2}};{\frac {\lambda \nu _{1}}{2\left(\nu _{1}+\nu _{2}\right)}}\right)}{B\left({\frac {\nu _{1}}{2}},{\frac {\nu _{2}}{2}}\right)}},\\[12pt]f'(1)={\frac {e^{-\lambda /2}\nu _{1}^{\frac {\nu _{1}}{2}}\nu _{2}^{\frac {\nu _{2}}{2}}\left(\nu _{1}+\nu _{2}\right){}^{{\frac {1}{2}}\left(-\nu _{1}-\nu _{2}-2\right)}\left(\nu _{2}\left(\lambda \,_{1}F_{1}\left({\frac {1}{2}}\left(\nu _{1}+\nu _{2}+2\right);{\frac {1}{2}}\left(\nu _{1}+2\right);{\frac {\lambda \nu _{1}}{2\left(\nu _{1}+\nu _{2}\right)}}\right)-2\,_{1}F_{1}\left({\frac {1}{2}}\left(\nu _{1}+\nu _{2}\right);{\frac {\nu _{1}}{2}};{\frac {\lambda \nu _{1}}{2\left(\nu _{1}+\nu _{2}\right)}}\right)\right)-2\nu _{1}\,_{1}F_{1}\left({\frac {1}{2}}\left(\nu _{1}+\nu _{2}\right);{\frac {\nu _{1}}{2}};{\frac {\lambda \nu _{1}}{2\left(\nu _{1}+\nu _{2}\right)}}\right)\right)}{2B\left({\frac {\nu _{1}}{2}},{\frac {\nu _{2}}{2}}\right)}}\end{array}}\right\}}$

## Special cases

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

## Related distributions

${\displaystyle Z=\lim _{\nu _{2}\to \infty }\nu _{1}F}$

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

1. ^ S. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, (New Jersey: Prentice Hall, 1998), p. 29.
2. ^ John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde. "Noncentral F Distribution: Boost 1.39.0". Boost.org. Retrieved 20 August 2011.
3. ^ Sigbert Klinke (10 December 2008). "Comparison of noncentral and central distributions". Humboldt-Universität zu Berlin.