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Noncentral chi-squared Distribution

The distribution of \(\sum_{i=1}^{\nu}\left(Z_{i}+\delta_{i}\right)^{2}\) where \(Z_{i}\) are independent standard normal variables and \(\delta_{i}\) are constants. \(\lambda=\sum_{i=1}^{\nu}\delta_{i}^{2}>0.\) (In communications it is called the Marcum-Q function). It can be thought of as a Generalized Rayleigh-Rice distribution.

The two shape parameters are \(\nu\), a positive integer, and \(\lambda\), a positive real number. The support is \(x\geq0\).

\begin{eqnarray*} f\left(x;\nu,\lambda\right) & = & e^{-\left(\lambda+x\right)/2}\frac{1}{2}\left(\frac{x}{\lambda}\right)^{\left(\nu-2\right)/4}I_{\left(\nu-2\right)/2}\left(\sqrt{\lambda x}\right)\\ F\left(x;\nu,\lambda\right) & = & \sum_{j=0}^{\infty}\left\{ \frac{\left(\lambda/2\right)^{j}}{j!}e^{-\lambda/2}\right\} \mathrm{Pr}\left[\chi_{\nu+2j}^{2}\leq x\right]\\ G\left(q;\nu,\lambda\right) & = & F^{-1}\left(q;\nu,\lambda\right)\\ \mu & = & \nu+\lambda\\ \mu_{2} & = & 2\left(\nu+2\lambda\right)\\ \gamma_{1} & = & \frac{\sqrt{8}\left(\nu+3\lambda\right)}{\left(\nu+2\lambda\right)^{3/2}}\\ \gamma_{2} & = & \frac{12\left(\nu+4\lambda\right)}{\left(\nu+2\lambda\right)^{2}}\end{eqnarray*}

where \(I_{\nu }(y)\) is a modified Bessel function of the first kind.

References

• “Noncentral chi-squared distribution”, Wikipedia https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

Implementation: scipy.stats.ncx2