F-Distribution

Arises in the testing of whether two observed samples have the same Variance. Let ${\chi_m}^2$ and ${\chi_n}^2$ be independent variates distributed as Chi-Squared with $m$ and $n$ Degrees of Freedom. Define a statistic $F_{n,m}$ as the ratio of the dispersions of the two distributions

$$F_{n,m} \equiv {{\chi_n}^2/n\over {\chi_m}^2/m}.$$ (1)

This statistic then has an $F$-distribution with probability function and cumulative distribution

$$F_{n,m}(x) = {\Gamma({\textstyle{n+m\over 2}})n^{n/2}m^{m/2}\over\Gamma({\textstyle{n\over 2}})\Gamma({\textstyle{m\over 2}})}{x^{n/2-1}\over (m+nx)^{(n+m)/2}}$$ (2)

$$= {m^{m/2}n^{n/2} x^{n/2-1}\over (m+nx)^{(n+m)/2} B({\textstyle{1\over 2}}n,{\textstyle{1\over 2}}m)}$$ (3)

$$= I(1;{\textstyle{1\over 2}}m;{\textstyle{1\over 2}}n)-I\left({{m\over m+nx}; {\textstyle{1\over 2}}m; {\textstyle{1\over 2}}n}\right),$$ (4)

where $\Gamma(z)$ is the Gamma Function, $B(a,b)$ is the Beta Function, and $I(a,b;x)$ is the Regularized Beta Function. The Mean, Variance, Skewness and Kurtosis are

$$\mu = {m\over m-2}$$ (5)

$$\sigma^2 = {2m^2(m+n-2)\over n(m-2)^2(m-4)}$$ (6)

$$\gamma_1 = {2(m+2n-2)\over m-6} \sqrt{2(m-4)\over n(m+n-2)}$$ (7)

$$\gamma_2 = {12(-16+20m-8m^2+m^3+44n)\over n(m-6)(m-8)(n+m-2)}$$

$$\phantom{=} +{12(-32mn+5m^2n-22n^2+5mn^2)\over n(m-6)(m-8)(n+m-2)}.$$ (8)

The probability that $F$ would be as large as it is if the first distribution has a smaller variance than the second is denoted $Q(F_{n,m})$.

The noncentral $F$-distribution is given by

$$P(x) = e^{-\lambda/2+(\lambda n_1 x)/[2(n_2+n_1x)]} {n_1}^{n_1/2} {n_2}^{n_2/2} x^{n_1/2-1} (n_2+n_1x)^{-(n_1+n_2)/2}$$

$$\phantom{=} \times {\Gamma({\textstyle{1\over 2}}n_1)\Gamma(1+{\textstyle{1\o... ...ver 2}}n_1, {\textstyle{1\over 2}}n_2)\Gamma[{\textstyle{1\over 2}}(n_1+n_2)]},$$ (9)

where $\Gamma(z)$ is the Gamma Function, $B(\alpha,\beta)$ is the Beta Function, and $L_m^n(z)$ is an associated Laguerre Polynomial.

See also Beta Function, Gamma Function, Regularized Beta Function, Snedecor's F-Distribution

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 946-949, 1972.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Incomplete Beta Function, Student's Distribution, F-Distribution, Cumulative Binomial Distribution.'' §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 219-223, 1992.

Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 117-118, 1992.