Noncentral F-distribution

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.

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

$F=\frac{X/\nu_1}{Y/\nu_2}$

is a noncentral F-distributed random variable. The probability density function (pdf) for the noncentral F-distribution is^[1]

$p(f) =\sum\limits_{k=0}^\infty\frac{e^{-\lambda/2}(\lambda/2)^k}{ B\left(\frac{\nu_2}{2},\frac{\nu_1}{2}+k\right) k!} \left(\frac{\nu_1}{\nu_2}\right)^{\frac{\nu_1}{2}+k} \left(\frac{\nu_2}{\nu_2+\nu_1f}\right)^{\frac{\nu_1+\nu_2}{2}+k}f^{\nu_1/2-1+k}$

when $f\ge0$ and zero otherwise. The degrees of freedom $\nu_1$ and $\nu_2$ are positive. The noncentrality parameter $\lambda$ is nonnegative. The term $B(x,y)$ is the beta function, where

$B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}.$

The cumulative distribution function for the noncentral F-distribution is

$F(x|d_1,d_2,\lambda)=\sum\limits_{j=0}^\infty\left(\frac{\left(\frac{1}{2}\lambda\right)^j}{j!}e^{-\frac{\lambda}{2}}\right)I\left(\frac{d_1x}{d_2 + d_1x}\bigg|\frac{d_1}{2}+j,\frac{d_2}{2}\right)$

where $I$ is the regularized incomplete beta function.

The mean and variance of the noncentral F-distribution are

$\operatorname{E}\left[F\right]= \begin{cases} \frac{\nu_2(\nu_1+\lambda)}{\nu_1(\nu_2-2)} &\nu_2>2\\ \text{Does not exist} &\nu_2\le2\\ \end{cases}$

and

$\operatorname{Var}\left[F\right]= \begin{cases} 2\frac{(\nu_1+\lambda)^2+(\nu_1+2\lambda)(\nu_2-2)}{(\nu_2-2)^2(\nu_2-4)}\left(\frac{\nu_2}{\nu_1}\right)^2 &\nu_2>4\\ \text{Does not exist} &\nu_2\le4.\\ \end{cases}$

Differential equation

The pdf of the noncentral F-distribution is a solution of the following differential equation:

$\left\{\begin{array}{l} 4 x \left(\nu _2+\nu_1 x\right){}^2 f''(x)+f'(x) \left(-2 \nu _2^2 \nu _1+8 \nu _2^2+ 16 \nu _1^2 x^2+4 \nu_2 \nu_1^2 x^2-2 \lambda \nu_2 \nu _1 x-2 \nu_2 \nu_1^2 x+4 \nu_2^2 \nu_1 x+24 \nu_2 \nu_1 x\right)+\nu_1 \left(\nu_2+2\right) f(x) \left(-\lambda \nu_2-\nu_2 \nu_1+4 \nu_2+4 \nu_1 x+\nu_2 \nu_1 x\right)=0, \\[12pt] f(1)=\frac{e^{-\lambda /2} \nu_1^{\frac{\nu_1}{2}} \nu_2^{\frac{\nu _2}{2}} \left(\nu _1+\nu _2\right){}^{\frac{1}{2} \left(-\nu _1-\nu _2\right)} \, _1F_1\left(\frac{1}{2} \left(\nu _1+\nu _2\right);\frac{\nu_1}{2}; \frac{\lambda \nu _1}{2 \left(\nu _1+\nu _2\right)}\right)}{B\left(\frac{\nu_1}{2}, \frac{\nu_2}{2}\right)}, \\[12pt] f'(1)=\frac{e^{-\lambda/2} \nu_1^{\frac{\nu_1}{2}} \nu_2^{\frac{\nu_2}{2}} \left(\nu _1+\nu _2\right){}^{\frac{1}{2} \left(-\nu_1-\nu_2-2\right)} \left(\nu_2 \left(\lambda \, _1F_1\left(\frac{1}{2} \left(\nu_1+\nu_2+2\right); \frac{1}{2} \left(\nu_1+2\right);\frac{\lambda \nu_1}{2 \left(\nu_1+\nu_2\right)}\right)-2 \, _1F_1\left(\frac{1}{2} \left(\nu_1+\nu_2\right);\frac{\nu_1}{2}; \frac{\lambda\nu_1}{2 \left(\nu_1+\nu _2\right)}\right)\right)-2 \nu _1 \, _1F_1\left(\frac{1}{2} \left(\nu_1+\nu_2\right);\frac{\nu_1}{2}; \frac{\lambda \nu_1}{2 \left(\nu_1+\nu_2\right)}\right)\right)} {2 B\left(\frac{\nu_1}{2},\frac{\nu_2}{2}\right)} \end{array}\right\}$

Special cases

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

Related distributions

Z has a noncentral chi-squared distribution if

$Z=\lim_{\nu_2\to\infty}\nu_1 F$

where F has a noncentral F-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. John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde. "Noncentral F Distribution: Boost 1.39.0". Boost.org. Retrieved 20 August 2011.
3. Sigbert Klinke (10 December 2008). "Comparison of noncentral and central distributions". Humboldt-Universität zu Berlin.

References

• Weisstein, Eric W.; et al. "Noncentral F-distribution". MathWorld. Wolfram Research, Inc. Retrieved 20 August 2011.

External links