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''/''n''₁)/(''Y''/''n''₂), where the numerator ''X'' has a noncentral chi-squared distribution with ''n''₁ degrees of freedom and the denominator ''Y'' has a central chi-squared distribution with ''n''₂ 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
:$F=\frac$
is a noncentral ''F''-distributed random variable.
The probability density function (pdf) for the noncentral ''F''-distribution is
:$p(f) =\sum\limits_^\infty\frac \left(\frac\right)^ \left(\frac\right)^f^$
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
:$B(x,y)=\frac.$

The cumulative distribution function for the noncentral ''F''-distribution is
:$F(x\mid d_1,d_2,\lambda)=\sum\limits_^\infty\left(\frace^ \right)I\left(\frac\bigg, \frac+j,\frac\right)$
where $I$ is the regularized incomplete beta function.

The mean and variance of the noncentral ''F''-distribution are
:
\operatorname \quad
\begin
= \frac & \text \nu_2>2\\
\text & \text \nu_2\le2\\
\end

and
:
\operatorname \quad
\begin
= 2\frac\left(\frac\right)^2
& \text \nu_2>4\\
\text
& \text \nu_2\le4.\\
\end

Special cases

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

Related distributions

''Z'' has a noncentral chi-squared distribution if

: $Z=\lim_\nu_1 F$

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.

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.

Notes

References

*

{{Probability distributions

Continuous distributions
F