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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