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.
Occurrence and specification
If is a noncentral chi-squared random variable with noncentrality parameter and degrees of freedom, and is a chi-squared random variable with degrees of freedom that is statistically independent of , then
when and zero otherwise. The degrees of freedom and are positive. The noncentrality parameter is nonnegative. The term is the beta function, where
The cumulative distribution function for the noncentral F-distribution is
where is the regularized incomplete beta function.
The mean and variance of the noncentral F-distribution are
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The pdf of the noncentral F-distribution is a solution of the following differential equation:
When λ = 0, the noncentral F-distribution becomes the F-distribution.
Z has a noncentral chi-squared distribution if
where F has a noncentral F-distribution.
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.
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