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

$p(f)=\sum _{k=0}^{\mathrm{\infty }}\frac{e^{-\lambda /2}(\lambda /2{)}^k}{B(\frac{{\nu }_2}{2},\frac{{\nu }_1}{2}+k)k!}{\left(\frac{{\nu }_1}{{\nu }_2}\right)}^{\frac{{\nu }_1}{2}+k}{\left(\frac{{\nu }_2}{{\nu }_2+{\nu }_{1}f}\right)}^{\frac{{\nu }_1+{\nu }_2}{2}+k}{f}^{{\nu }_1/2-1+k}$

when $f\ge 0$ and zero otherwise. The degrees of freedom ${\nu }_1$ and ${\nu }_2$ are positive. The term $B(x,y)$ is the beta function, where

$B(x,y)=\frac{\mathrm{\Gamma }(x)\mathrm{\Gamma }(y)}{\mathrm{\Gamma }(x+y)}.$

The cumulative distribution function for the noncentral F-distribution is

$F(x\mid d_1,d_2,\lambda )=\sum _{j=0}^{\mathrm{\infty }}\left(\frac{{\left(\frac{1}{2}\lambda \right)}^j}{j!}{e}^{-\lambda /2}\right)I\left(\frac{d_{1}x}{d_2+d_{1}x}\right|\frac{d_1}{2}+j,\frac{d_2}{2})$

where $I$ is the regularized incomplete beta function. The mean and variance of the noncentral F-distribution are

$\mathrm{E}[F]\{\begin{array}{ll}=\frac{{\nu }_2({\nu }_1+\lambda )}{{\nu }_1({\nu }_2-2)}& \text{if }{\nu }_2>2\\ \text{does not exist}& \text{if }{\nu }_2\le 2\\ \end{array}$

and

$\mathrm{Var}[F]\{\begin{array}{ll}=2\frac{({\nu }_1+\lambda {)}^2+({\nu }_1+2\lambda )({\nu }_2-2)}{({\nu }_2-2{)}^2({\nu }_2-4)}{\left(\frac{{\nu }_2}{{\nu }_1}\right)}^2& \text{if }{\nu }_2>4\\ \text{does not exist}& \text{if }{\nu }_2\le 4.\\ \end{array}$

Special cases

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

Related distributions

Z has a noncentral chi-squared distribution if

$Z=\underset{{\nu }_2\to \mathrm{\infty }}{\lim }{\nu }_{1}F$

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 of the Humboldt University of Berlin.

References

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