Noncentral F-distribution explained

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 λ 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 , thenF = (X/\nu _{1})/(Y/\nu _{2}) is a noncentral F-distributed random variable.The probability density function (pdf) for the noncentral F-distribution is^[1] p(f) = \sum\limits ^{inf ty (e ^{-λ/2} ( λ / 2 ) ^{k})/(B\left( (\nu _{2})/(2) , (\nu _{1})/(2) + k\right) k!)} _{k =0} \left( (\nu _{1})/(\nu _{2}) \right) ^{(\nu _{1})/(+ k 2)} \left( (\nu _{2})/(\nu _{2+\nu} _{1f}) \right) ^{(\nu _{1+\nu} _{2})/(+ k 2)} 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, whereB(x,y) = (\Gamma(x)\Gamma(y))/(\Gamma(x + y)) .

The cumulative distribution function for the noncentral F-distribution is

F(x\mid d _{1,d} _{2,λ)=\sum\limits} ^{inf ty\left( (\left( (1)/(λ \right) ^{j} 2))/(j!)} ^{j =0} e ^{-λ/2} \right)I\left( (d _{1x})/(| d _{2} + d _{1x}) (d _{1})/(+ j, 2) (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] \begin{cases} = (\nu _{2(\nu} _{1+λ)})/(\nu _{1(\nu} _{2-2)}) & if \nu _{2>2\\ doesnotexist} & if \nu _{2\le2\\ \end{cases}} and\operatorname{Var}[F] \begin{cases} =2 ((\nu ^{2+ (\nu} _{1+2λ )(\nu} _{2-2)})/(\left( (\nu ^{2 (\nu} _{2-4 )}) (\nu _{2})/(\nu _{1}) \right) ^{2 &} if \nu _{2>4\\ doesnotexist &} if \nu _{2\le4.\\ \end{cases}}

Special cases

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

Related distributions

Z has a noncentral chi-squared distribution if

Z = \lim _{\nu _{2\toinfty}} \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.^[2]

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.^[3]

Notes

1. S. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, (New Jersey: Prentice Hall, 1998), p. 29.
2. Web site: Noncentral F Distribution: Boost 1.39.0 . John Maddock . Paul A. Bristow . Hubert Holin . Xiaogang Zhang . Bruno Lalande . Johan Råde . Boost.org . 20 August 2011.
3. Web site: Comparison of noncentral and central distributions . Sigbert Klinke . 10 December 2008 . Humboldt-Universität zu Berlin.

References

• Web site: Noncentral F-distribution . Eric W.. Weisstein . Eric W. Weisstein . . Wolfram Research, Inc . 20 August 2011. etal.