.. _continuous-ncf:

Noncentral F Distribution

The distribution of :math:\left(X_{1}/X_{2}\right)\left(\nu_{2}/\nu_{1}\right) if :math:X_{1} is non-central chi-squared with :math:\nu_{1} degrees of freedom and parameter :math:\lambda, and :math:X_{2} is chi-squared with :math:\nu_{2} degrees of freedom.

There are 3 shape parameters: the degrees of freedom :math:\nu_{1}>0 and :math:\nu_{2}>0; and :math:\lambda\geq 0.

.. math:: :nowrap:

\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 :math:L_{\nu_{2}/2}^{\nu_{1}/2-1}(x) is an associated Laguerre polynomial.

If :math:\lambda=0, the distribution becomes equivalent to the Fisher distribution with :math:\nu_{1} and :math:\nu_{2} degrees of freedom.

Implementation: scipy.stats.ncf