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 <math>X</math> is a noncentral chi-squared random variable with noncentrality parameter <math>\lambda</math> and <math>\nu_1</math> degrees of freedom, and <math>Y</math> is a chi-squared random variable with <math>\nu_2</math> degrees of freedom that is statistically independent of <math>X</math>, then

<math>

F=\frac{X/\nu_1}{Y/\nu_2} </math> is a noncentral F-distributed random variable. The probability density function (pdf) for the noncentral F-distribution is<ref>Template:Cite book</ref>

<math>

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}

</math>

when <math>f\ge0</math> and zero otherwise. The degrees of freedom <math>\nu_1</math> and <math>\nu_2</math> are positive. The term <math>B(x,y)</math> is the beta function, where

<math>

B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}. </math>

The cumulative distribution function for the noncentral F-distribution is

<math>

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) </math> where <math>I</math> is the regularized incomplete beta function.

The mean and variance of the noncentral F-distribution are

<math>

\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} </math> and

<math>

\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} </math>

Special cases

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

Related distributions

Z has a noncentral chi-squared distribution if

<math> Z=\lim_{\nu_2\to\infty}\nu_1 F </math>

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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Notes

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References

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