Noncentral F-distribution - Occurrence and Specification

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

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

is a noncentral F-distributed random variable. The probability density function for the noncentral F-distribution is

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 and zero otherwise. The degrees of freedom and are positive. The noncentrailty parameter is nonnegative. The term 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_1F}{d_2 + d_1F}\bigg|\frac{d_1}{2}+j,\frac{d_2}{2}\right)

where is the regularized incomplete beta function.


The mean and variance of the noncentral F-distribution are

\mbox{E}\left= \begin{cases} \frac{\nu_2(\nu_1+\lambda)}{\nu_1(\nu_2-2)} &\nu_2>2\\ \mbox{Does not exist} &\nu_2\le2\\ \end{cases}

and

\mbox{Var}\left= \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\\ \mbox{Does not exist} &\nu_2\le4.\\ \end{cases}

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