Nakagami Distribution

Generalization of the chi distribution. Shape parameter is \(\nu>0.\) Defined for \(x>0.\)

\begin{eqnarray*} f\left(x;\nu\right) & = & \frac{2\nu^{\nu}}{\Gamma\left(\nu\right)}x^{2\nu-1}\exp\left(-\nu x^{2}\right)\\ F\left(x;\nu\right) & = & \Gamma\left(\nu,\nu x^{2}\right)\\ G\left(q;\nu\right) & = & \sqrt{\frac{1}{\nu}\Gamma^{-1}\left(v,q\right)}\end{eqnarray*}

\begin{eqnarray*} \mu & = & \frac{\Gamma\left(\nu+\frac{1}{2}\right)}{\sqrt{\nu}\Gamma\left(\nu\right)}\\ \mu_{2} & = & \left[1-\mu^{2}\right]\\ \gamma_{1} & = & \frac{\mu\left(1-4v\mu_{2}\right)}{2\nu\mu_{2}^{3/2}}\\ \gamma_{2} & = & \frac{-6\mu^{4}\nu+\left(8\nu-2\right)\mu^{2}-2\nu+1}{\nu\mu_{2}^{2}}\end{eqnarray*}

Implementation: scipy.stats.nakagami