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Nakagami Distribution

Generalization of the chi distribution. Shape parameter is \(\nu>0.\) The support is \(x\geq0.\)

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

where \(\gamma\) is the lower incomplete gamma function, \(\gamma\left(\nu, x\right) = \int_0^x t^{\nu-1} e^{-t} dt\).

\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