Noncentral F Distribution#
The distribution of \(\left(X_{1}/X_{2}\right)\left(\nu_{2}/\nu_{1}\right)\) if \(X_{1}\) is non-central chi-squared with \(\nu_{1}\) degrees of freedom and parameter \(\lambda\), and \(X_{2}\) is chi-squared with \(\nu_{2}\) degrees of freedom.
There are 3 shape parameters: the degrees of freedom \(\nu_{1}>0\) and \(\nu_{2}>0\); and \(\lambda\geq 0\).
\begin{eqnarray*}
f\left(x;\lambda,\nu_{1},\nu_{2}\right)
& = &
\exp\left[\frac{\lambda}{2} +
\frac{\left(\lambda\nu_{1}x\right)}
{2\left(\nu_{1}x+\nu_{2}\right)}
\right]
\nu_{1}^{\nu_{1}/2}\nu_{2}^{\nu_{2}/2}x^{\nu_{1}/2-1} \\
& &
\times\left(\nu_{2}+\nu_{1}x\right)^{-\left(\nu_{1}+\nu_{2}\right)/2}
\frac{\Gamma\left(\frac{\nu_{1}}{2}\right)
\Gamma\left(1+\frac{\nu_{2}}{2}\right)
L_{\nu_{2}/2}^{\nu_{1}/2-1}
\left(-\frac{\lambda\nu_{1}x}
{2\left(\nu_{1}x+\nu_{2}\right)}\right)}
{B\left(\frac{\nu_{1}}{2},\frac{\nu_{2}}{2}\right)
\Gamma\left(\frac{\nu_{1}+\nu_{2}}{2}\right)}
\end{eqnarray*}
where \(L_{\nu_{2}/2}^{\nu_{1}/2-1}(x)\) is an associated Laguerre polynomial.
If \(\lambda=0\), the distribution becomes equivalent to the Fisher distribution with \(\nu_{1}\) and \(\nu_{2}\) degrees of freedom.
Implementation: scipy.stats.ncf