Indefinite Integration of the Gamma Integral
and Related Statistical Applications

4.7 Marcum Q-function

The cumulative function of the noncentral chi-square function has a form of the Marcum Q-function

$$\begin{equation*} 1-{{Q}_{\tfrac{k}{2}}}\left( \sqrt{\lambda },\sqrt{x} \right), \end{equation*}$$
where $k>0$ (degree of freedom), $\lambda >0$ (noncentrailty parameter), and $x\in \left[ 0,\infty \right)$.
The Marcum Q-function (Nuttall, 1975) is defined as
$$\begin{equation*} {{Q}_{M}}\left( a,b \right)={{\int_{b}^{\infty }{x\left( \frac{x}{a} \right)}}^{M-1}}\exp \left( \frac{-\left( {{x}^{2}}+{{a}^{2}} \right)}{2} \right){{I}_{M-1}}\left( ax \right)dx, \end{equation*}$$
where ${{I}_{M-1}}\left( ax \right)$ is the modified Bessel function of the first kind of ($M-1$)th order, and it is defined as
$$\begin{equation*} {{I}_{M-1}}\left( ax \right)=\frac{1}{\Gamma \left( M \right)}{{\left( \frac{ax}{2} \right)}^{M-1}}\sum\limits_{i=0}^{\infty}{\frac{1}{i!\prod\limits_{j=1}^{i}{\left( M-1+j \right)}}}{{\left( \frac{ax}{2} \right)}^{2i}}. \end{equation*}$$

With some manipulations, we can simplify the Marcum Q-function into an "$h$" function

$$\begin{equation*} {{Q}_{M}}\left(a,b\right)=\frac{\exp\left(\frac{-{{a}^{2}}}{2} \right)}{{{2}^{M}}\Gamma\left(M\right)}\sum\limits_{i=0}^{\infty}{{{\left( \frac{a}{2} \right)}^{2i}}}\frac{{{C}^{M+i}}{\exp({\tfrac{-C}{2}})}h_{M-1+i}^{\tfrac{-C}{2}}-{{b}^{2\left(M+i \right)}}{\exp({\tfrac{-{{b}^{2}}}{2}})}h_{M-1+i}^{\tfrac{-{{b}^{2}}}{2}}}{i!\prod\limits_{j=1}^{i}{\left( M-1+j \right)}}. \tag{4.12} \end{equation*}$$
[Proof for (4.12)]

In addition, the Marcum Q-function also applies to the cumulative distribution function of the Rice distribution as shown in the supplementary material.

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