Noncentral F-distribution


In probability theory and statistics, the noncentral F-distribution is a continuous probability distribution that is a generalization of the F-distribution. It describes the distribution of the quotient /, 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 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
is a noncentral F-distributed random variable.
The probability density function for the noncentral F-distribution is
when and zero otherwise.
The degrees of freedom and are positive.
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
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, in MATLAB in Mathematica, in NumPy, 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.