The noncentral f-ratio distribution is a continuous probability distribution with probability density function given by:

subject to the following conditions:

The NonCentralFRatio variate with degrees of freedom nu and omega and noncentrality parameter delta=0 is equivalent to the FRatio variate with degrees of freedom nu and omega.

The NonCentralFRatio variate with degrees of freedom nu and omega and noncentrality parameter delta is related to the independent NonCentralChiSquare variate and ChiSquare variate by NonCentralFRatio(nu,omega,delta) ~ (NonCentralChiSquare(nu,delta)*omega)/(ChiSquare(omega)*nu)

Note that the NonCentralFRatio command is inert and should be used in combination with the RandomVariable command.

The Quantile and CDF functions applied to a noncentral F-ratio distribution use a sequence of iterations in order to converge on the desired output point.  The maximum number of iterations to perform is equal to 100 by default, but this value can be changed by setting the environment variable _EnvStatisticsIterations to the desired number of iterations.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{NonCentralFRatio}\left(\mathrm{\nu }\,\mathrm{\omega }\,\mathrm{\delta }\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\left\{\begin{array}{cc}0& u<0\\ \frac{\left(1+\frac{u\mathrm{\delta }\mathrm{hypergeom}\left(\left[1\&comma;\frac{\mathrm{\nu }}{2}+\frac{\mathrm{\omega }}{2}+1\right]\&comma;\left[2\&comma;\frac{\mathrm{\nu }}{2}+1\right]\&comma;\frac{\mathrm{\delta }u\mathrm{\nu }}{2\left(\mathrm{\nu }u+\mathrm{\omega }\right)}\right)\left(\mathrm{\nu }+\mathrm{\omega }\right)}{2\left(\mathrm{\nu }u+\mathrm{\omega }\right)}\right){\&ExponentialE;}^{-\frac{\mathrm{\delta }}{2}}{\mathrm{\nu }}^{\frac{\mathrm{\nu }}{2}}{\mathrm{\omega }}^{\frac{\mathrm{\omega }}{2}}{u}^{\frac{\mathrm{\nu }}{2}-1}}{\mathrm{{\rm B}}\left(\frac{\mathrm{\nu }}{2}\&comma;\frac{\mathrm{\omega }}{2}\right){\left(\mathrm{\nu }u+\mathrm{\omega }\right)}^{\frac{\mathrm{\nu }}{2}+\frac{\mathrm{\omega }}{2}}}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{PDF}\left(X\&comma;0.5\right)$

$\frac{\left(1.+\frac{0.2500000000\mathrm{\delta }\mathrm{hypergeom}\left(\left[1.\&comma;0.5000000000\mathrm{\nu }+0.5000000000\mathrm{\omega }+1.\right]\&comma;\left[2.\&comma;0.5000000000\mathrm{\nu }+1.\right]\&comma;\frac{0.2500000000\mathrm{\delta }\mathrm{\nu }}{\mathrm{\omega }+0.5\mathrm{\nu }}\right)\left(\mathrm{\nu }+\mathrm{\omega }\right)}{\mathrm{\omega }+0.5\mathrm{\nu }}\right){\&ExponentialE;}^{-0.5000000000\mathrm{\delta }}{\mathrm{\nu }}^{0.5000000000\mathrm{\nu }}{\mathrm{\omega }}^{0.5000000000\mathrm{\omega }}{0.5}^{0.5000000000\mathrm{\nu }-1.}}{\mathrm{{\rm B}}\left(0.5000000000\mathrm{\nu }\&comma;0.5000000000\mathrm{\omega }\right){\left(\mathrm{\omega }+0.5\mathrm{\nu }\right)}^{0.5000000000\mathrm{\nu }+0.5000000000\mathrm{\omega }}}$

$\mathrm{Mean}\left(X\right)$

$\left\{\begin{array}{cc}\mathrm{undefined}& \mathrm{\omega }\le 2\\ \frac{\mathrm{\omega }\left(\mathrm{\nu }+\mathrm{\delta }\right)}{\mathrm{\nu }\left(\mathrm{\omega }-2\right)}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{Variance}\left(X\right)$

$\left\{\begin{array}{cc}\mathrm{undefined}& \mathrm{\omega }\le 4\\ \frac{2{\mathrm{\omega }}^2\left({\left(\mathrm{\nu }+\mathrm{\delta }\right)}^2+\left(\mathrm{\nu }+2\mathrm{\delta }\right)\left(\mathrm{\omega }-2\right)\right)}{{\mathrm{\nu }}^2{\left(\mathrm{\omega }-2\right)}^2\left(\mathrm{\omega }-4\right)}& \mathrm{otherwise}\end{array}\right.$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.