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 ${\displaystyle X}$ 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 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

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

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

The mean and variance of the noncentral F-distribution are

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

and

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

Special cases

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

Related distributions

Z has a noncentral chi-squared distribution if

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

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

1. Kay, S. (1998). Fundamentals of Statistical Signal Processing: Detection Theory. New Jersey: Prentice Hall. p. 29. ISBN 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.
3. Sigbert Klinke (10 December 2008). "Comparison of noncentral and central distributions". Humboldt-Universität zu Berlin.

References

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