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 $$X$$ 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_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} $$

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.

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, School of Business and Economics, Humboldt-Universität zu Berlin.