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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 is¹

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.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 of the Humboldt University of Berlin.³

Notes

• Weisstein, Eric W.; et al. "Noncentral F-distribution". MathWorld. Wolfram Research, Inc. Retrieved 20 August 2011.

References

1. Kay, S. (1998). Fundamentals of Statistical Signal Processing: Detection Theory. New Jersey: Prentice Hall. p. 29. ISBN 0-13-504135-X. 0-13-504135-X ↩

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. 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 ↩

3. Sigbert Klinke (10 December 2008). "Comparison of noncentral and central distributions". Humboldt-Universität zu Berlin. http://mars.wiwi.hu-berlin.de/mediawiki/slides/index.php/Comparison_of_noncentral_and_central_distributions ↩