Marcum Q-function

In statistics, the Marcum-Q-function ${\displaystyle Q_{M}}$ is defined as

${\displaystyle Q_{M}(a,b)=\int _{b}^{\infty }x\left({\frac {x}{a}}\right)^{M-1}\exp \left(-{\frac {x^{2}+a^{2}}{2}}\right)I_{M-1}(ax)\,dx}$

or as

${\displaystyle Q_{M}(a,b)=\exp \left(-{\frac {a^{2}+b^{2}}{2}}\right)\sum _{k=1-M}^{\infty }\left({\frac {a}{b}}\right)^{k}I_{k}(ab)}$

with modified Bessel function ${\displaystyle I_{M-1}}$ of order M − 1. The Marcum Q-function is used for example as a cumulative distribution function (more precisely, as a survivor function) for noncentral chi, noncentral chi-squared and Rice distributions.

For non-integer values of M, the Marcum Q function can be defined as[1]

${\displaystyle {\begin{aligned}Q_{M}(a,b)&=1-e^{-a^{2}/2}\sum _{k=0}^{\infty }\left({\frac {a^{2}}{2}}\right)^{k}{\frac {\gamma (M+k,{\frac {b^{2}}{2}})}{k!\Gamma (M+k)}}\\[6pt]&=1-e^{-a^{2}/2}\sum _{k=0}^{\infty }\left({\frac {a^{2}}{2}}\right)^{k}{\frac {P(M+k,{\frac {b^{2}}{2}})}{k!}}\end{aligned}}}$

where ${\displaystyle P(s,x)}$ is the regularized Gamma function.

The Marcum Q-function is monotonic and log-concave.[2]