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 is a noncentral F-distributed random variable. The probability density function (pdf) for the noncentral F-distribution is when f\ge0 and zero otherwise. The degrees of freedom \nu_1 and \nu_2 are positive. The term B(x,y) is the beta function, where The cumulative distribution function for the noncentral F-distribution is where I is the regularized incomplete beta function. The mean and variance of the noncentral F-distribution are and

Special cases

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

Related distributions

Z has a noncentral chi-squared distribution if 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.