Noncentral F-distribution

Occurrence and specification

If X is a noncentral chi-squared random variable with noncentrality parameter \lambda and \nu_1 degrees of freedom, and Y is a chi-squared random variable with \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 : 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\ge0 and zero otherwise. The degrees of freedom \nu_1 and \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_22\ \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_24\ \text{does not exist} & \text{if } \nu_2\le4.\ \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_{\nu_2\to\infty}\nu_1 F

where F has a noncentral F-distribution.

See also noncentral t-distribution.

A Doubly noncentral F distribution has a noncentral chi-squared distribution in the numerator and denominator.

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.

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.

Notes

References

References

1. Kay, S.. (1998). "Fundamentals of Statistical Signal Processing: Detection Theory". Prentice Hall.
2. Leemis, Larry. "Doubly noncentral ''F''-distribution".
3. John Maddock. "Noncentral F Distribution: Boost 1.39.0". Boost.org.
4. Sigbert Klinke. (10 December 2008). "Comparison of noncentral and central distributions". Humboldt-Universität zu Berlin.