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 $X$ 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 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 noncentrality parameter $\lambda$ is nonnegative. 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

$F(x|d_1,d_2,\lambda)=\sum\limits_{j=0}^\infty\left(\frac{\left(\frac{1}{2}\lambda\right)^j}{j!}e^{-\frac{\lambda}{2}}\right)I\left(\frac{d_1x}{d_2 + d_1x}\bigg|\frac{d_1}{2}+j,\frac{d_2}{2}\right)$

where $I$ is the regularized incomplete beta function.

The mean and variance of the noncentral F-distribution are

$\mbox{E}\left[F\right]= \begin{cases} \frac{\nu_2(\nu_1+\lambda)}{\nu_1(\nu_2-2)} &\nu_2>2\\ \mbox{Does not exist} &\nu_2\le2\\ \end{cases}$

and

$\mbox{Var}\left[F\right]= \begin{cases} 2\frac{(\nu_1+\lambda)^2+(\nu_1+2\lambda)(\nu_2-2)}{(\nu_2-2)^2(\nu_2-4)}\left(\frac{\nu_2}{\nu_1}\right)^2 &\nu_2>4\\ \mbox{Does not exist} &\nu_2\le4.\\ \end{cases}$

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.

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 R language, for noncentral t, chisquare, and F, at the Institute of Statistics and Econometrics, School of Business and Economics, Humboldt-Universität zu Berlin.

References

• Weisstein, Eric W., et al Noncentral F-distribution. MathWorld. Wolfram Research, Inc. URL accessed on 20 August 2011.