Marcum Q-function

In statistics, the generalized Marcum Q-function of order $\nu$ is defined as

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

where $b\geq 0$ and $a,\nu >0$ and $I_{\nu -1}$ is the modified Bessel function of first kind of order $\nu -1$. If $b>0$, the integral converges for any $\nu$. The Marcum Q-function occurs as a complementary cumulative distribution function for noncentral chi, noncentral chi-squared, and Rice distributions. In engineering, this function appears in the study of radar systems, communication systems, queueing system, and signal processing. This function was first studied for $\nu =1$, and hence named after, by Jess Marcum for pulsed radars.

Properties

Finite integral representation

Using the fact that $Q_{\nu }(a,0)=1$, the generalized Marcum Q-function can alternatively be defined as a finite integral as

$Q_{\nu }(a,b)=1-{\frac {1}{a^{\nu -1}}}\int _{0}^{b}x^{\nu }\exp \left(-{\frac {x^{2}+a^{2}}{2}}\right)I_{\nu -1}(ax)\,dx.$

However, it is preferable to have an integral representation of the Marcum Q-function such that (i) the limits of the integral are independent of the arguments of the function, (ii) and that the limits are finite, (iii) and that the integrand is a Gaussian function of these arguments. For positive integer values of $\nu =n$, such a representation is given by the trigonometric integral

$Q_{n}(a,b)=\left\{{\begin{array}{lr}H_{n}(a,b)&a<b,\\{\frac {1}{2}}+H_{n}(a,a)&a=b,\\1+H_{n}(a,b)&a>b,\end{array}}\right.$

where

$H_{n}(a,b)={\frac {\zeta ^{1-n}}{2\pi }}\exp \left(-{\frac {a^{2}+b^{2}}{2}}\right)\int _{0}^{2\pi }{\frac {\cos(n-1)\theta -\zeta \cos n\theta }{1-2\zeta \cos \theta +\zeta ^{2}}}\exp(ab\cos \theta )\mathrm {d} \theta ,$

and the ratio $\zeta =a/b$ is a constant.

For any real $\nu >0$, such finite trigonometric integral is given by

$Q_{\nu }(a,b)=\left\{{\begin{array}{lr}H_{\nu }(a,b)+C_{\nu }(a,b)&a<b,\\{\frac {1}{2}}+H_{\nu }(a,a)+C_{\nu }(a,b)&a=b,\\1+H_{\nu }(a,b)+C_{\nu }(a,b)&a>b,\end{array}}\right.$

where $H_{n}(a,b)$ is as defined before, $\zeta =a/b$, and the additional correction term is given by

$C_{\nu }(a,b)={\frac {\sin(\nu \pi )}{\pi }}\exp \left(-{\frac {a^{2}+b^{2}}{2}}\right)\int _{0}^{1}{\frac {(x/\zeta )^{\nu -1}}{\zeta +x}}\exp \left[-{\frac {ab}{2}}\left(x+{\frac {1}{x}}\right)\right]\mathrm {d} x.$

For integer values of $\nu$, the correction term $C_{\nu }(a,b)$ tend to vanish.

Monotonicity and log-concavity

• The generalized Marcum Q-function $Q_{\nu }(a,b)$ is strictly increasing in $\nu$ and $a$ for all $a\geq 0$ and $b,\nu >0$, and is strictly decreasing in $b$ for all $a,b\geq 0$ and $\nu >0.$

• The function $\nu \mapsto Q_{\nu }(a,b)$ is log-concave on $[1,\infty )$ for all $a,b\geq 0.$

• The function $b\mapsto Q_{\nu }(a,b)$ is strictly log-concave on $(0,\infty )$ for all $a\geq 0$ and $\nu >1$, which implies that the generalized Marcum Q-function satisfies the new-is-better-than-used property.

• The function $a\mapsto 1-Q_{\nu }(a,b)$ is log-concave on $[0,\infty )$ for all $b,\nu >0.$

Series representation

• The generalized Marcum Q function of order $\nu >0$ can be represented using incomplete Gamma function as

$Q_{\nu }(a,b)=1-e^{-a^{2}/2}\sum _{k=0}^{\infty }{\frac {1}{k!}}{\frac {\gamma (\nu +k,{\frac {b^{2}}{2}})}{\Gamma (\nu +k)}}\left({\frac {a^{2}}{2}}\right)^{k},$

where $\gamma (s,x)$ is the lower incomplete Gamma function. This is usually called the canonical representation of the $\nu$-th order generalized Marcum Q-function.

• The generalized Marcum Q function of order $\nu >0$ can also be represented using generalized Laguerre polynomials as

$Q_{\nu }(a,b)=1-e^{-a^{2}/2}\sum _{k=0}^{\infty }(-1)^{k}{\frac {L_{k}^{(\nu -1)}({\frac {a^{2}}{2}})}{\Gamma (\nu +k+1)}}\left({\frac {b^{2}}{2}}\right)^{k+\nu },$

where $L_{k}^{(\alpha )}(\cdot )$ is the generalized Laguerre polynomial of degree $k$ and of order $\alpha$.

• The generalized Marcum Q-function of order $\nu >0$ can also be represented as Neumann series expansions

$Q_{\nu }(a,b)=e^{-(a^{2}+b^{2})/2}\sum _{\alpha =1-\nu }^{\infty }\left({\frac {a}{b}}\right)^{\alpha }I_{-\alpha }(ab),$

$1-Q_{\nu }(a,b)=e^{-(a^{2}+b^{2})/2}\sum _{\alpha =\nu }^{\infty }\left({\frac {b}{a}}\right)^{\alpha }I_{\alpha }(ab),$

where the summations are in increments of one. Note that when $\alpha$ assumes an integer value, we have $I_{\alpha }(ab)=I_{-\alpha }(ab)$.

• For non-negative half-integer values $\nu =n+1/2$, we have a closed form expression for the generalized Marcum Q-function as

$Q_{n+1/2}(a,b)={\frac {1}{2}}\left[\mathrm {erfc} \left({\frac {b-a}{\sqrt {2}}}\right)+\mathrm {erfc} \left({\frac {b+a}{\sqrt {2}}}\right)\right]+e^{-(a^{2}+b^{2})/2}\sum _{k=1}^{n}\left({\frac {b}{a}}\right)^{k-1/2}I_{k-1/2}(ab),$

where $\mathrm {erfc} (\cdot )$ is the complementary error function. Since Bessel functions with half-integer parameter have finite sum expansions as

$I_{\pm (n+0.5)}(z)={\frac {1}{\sqrt {\pi }}}\sum _{k=0}^{n}{\frac {(n+k)!}{k!(n-k)!}}\left[{\frac {(-1)^{k}e^{z}\mp (-1)^{n}e^{-z}}{(2z)^{k+0.5}}}\right],$

where $n$ is non-negative integer, we can exactly represent the generalized Marcum Q-function with half-integer parameter. More precisely, we have

$Q_{n+1/2}(a,b)=Q(b-a)+Q(b+a)+{\frac {1}{b{\sqrt {2\pi }}}}\sum _{i=1}^{n}\left({\frac {b}{a}}\right)^{i}\sum _{k=0}^{i-1}{\frac {(i+k-1)!}{k!(i-k-1)!}}\left[{\frac {(-1)^{k}e^{-(a-b)^{2}/2}+(-1)^{i}e^{-(a+b)^{2}/2}}{(2ab)^{k}}}\right],$

for non-negative integers $n$, where $Q(\cdot )$ is the Gaussian Q-function. Alternatively, we can also more compactly express the Bessel functions with half-integer as sum of hyperbolic sine and cosine functions:

$I_{n+{\frac {1}{2}}}(z)={\sqrt {\frac {2z}{\pi }}}\left[g_{n}(z)\sinh(z)+g_{-n-1}(z)\cosh(z)\right],$

where $g_{0}(z)=z^{-1}$, $g_{1}(z)=-z^{-2}$, and $g_{n-1}(z)-g_{n+1}(z)=(2n+1)z^{-1}g_{n}(z)$ for any integer value of $n$.

Recurrence relation and generating function

• Integrating by parts, we can show that generalized Marcum Q-function satisfies the following recurrence relation

$Q_{\nu +1}(a,b)-Q_{\nu }(a,b)=\left({\frac {b}{a}}\right)^{\nu }e^{-(a^{2}+b^{2})/2}I_{\nu }(ab).$

• The above formula is easily generalized as

$Q_{\nu -n}(a,b)=Q_{\nu }(a,b)-\left({\frac {b}{a}}\right)^{\nu }e^{-(a^{2}+b^{2})/2}\sum _{k=1}^{n}\left({\frac {a}{b}}\right)^{k}I_{\nu -k}(ab),$

$Q_{\nu +n}(a,b)=Q_{\nu }(a,b)+\left({\frac {b}{a}}\right)^{\nu }e^{-(a^{2}+b^{2})/2}\sum _{k=0}^{n-1}\left({\frac {b}{a}}\right)^{k}I_{\nu +k}(ab),$

for positive integer $n$. The former recurrence can be used to formally define the generalized Marcum Q-function for negative $\nu$. Taking $Q_{\infty }(a,b)=1$ and $Q_{-\infty }(a,b)=0$ for $n=\infty$, we obtain the Neumann series representation of the generalized Marcum Q-function.

• The related three-term recurrence relation is given by

$Q_{\nu +1}(a,b)-(1+c_{\nu }(a,b))Q_{\nu }(a,b)+c_{\nu }(a,b)Q_{\nu -1}(a,b)=0,$

where

$c_{\nu }(a,b)=\left({\frac {b}{a}}\right){\frac {I_{\nu }(ab)}{I_{\nu +1}(ab)}}.$

We can eliminate the occurrence of the Bessel function to give the third order recurrence relation

${\frac {a^{2}}{2}}Q_{\nu +2}(a,b)=\left({\frac {a^{2}}{2}}-\nu \right)Q_{\nu +1}(a,b)+\left({\frac {b^{2}}{2}}+\nu \right)Q_{\nu }(a,b)-{\frac {b^{2}}{2}}Q_{\nu -1}(a,b).$

• Another recurrence relationship, relating it with its derivatives, is given by

$Q_{\nu +1}(a,b)=Q_{\nu }(a,b)+{\frac {1}{a}}{\frac {\partial }{\partial a}}Q_{\nu }(a,b),$

$Q_{\nu -1}(a,b)=Q_{\nu }(a,b)+{\frac {1}{b}}{\frac {\partial }{\partial b}}Q_{\nu }(a,b).$

• The ordinary generating function of $Q_{\nu }(a,b)$ for integral $\nu$ is

$\sum _{n=-\infty }^{\infty }t^{n}Q_{n}(a,b)=e^{-(a^{2}+b^{2})/2}{\frac {t}{1-t}}e^{(b^{2}t+a^{2}/t)/2},$

where $|t|<1.$

Symmetry relation

• Using the two Neumann series representations, we can obtain the following symmetry relation for positive integral $\nu =n$

$Q_{n}(a,b)+Q_{n}(b,a)=1+e^{-(a^{2}+b^{2})/2}\left[I_{0}(ab)+\sum _{k=1}^{n-1}{\frac {a^{2k}+b^{2k}}{(ab)^{k}}}I_{k}(ab)\right].$

In particular, for $n=1$ we have

$Q_{1}(a,b)+Q_{1}(b,a)=1+e^{-(a^{2}+b^{2})/2}I_{0}(ab).$

Special values

Some specific values of Marcum-Q function are

• $Q_{\nu }(0,0)=1,$
• $Q_{\nu }(a,0)=1,$
• $Q_{\nu }(a,+\infty )=0,$
• $Q_{\nu }(0,b)={\frac {\Gamma (\nu ,b^{2}/2)}{\Gamma (\nu )}},$
• $Q_{\nu }(+\infty ,b)=1,$
• $Q_{\infty }(a,b)=1,$
• For $a=b$, by subtracting the two forms of Neumann series representations, we have

$Q_{1}(a,a)={\frac {1}{2}}[1+e^{-a^{2}}I_{0}(a^{2})],$

which when combined with the recursive formula gives

$Q_{n}(a,a)={\frac {1}{2}}[1+e^{-a^{2}}I_{0}(a^{2})]+e^{-a^{2}}\sum _{k=1}^{n-1}I_{k}(a^{2}),$

$Q_{-n}(a,a)={\frac {1}{2}}[1+e^{-a^{2}}I_{0}(a^{2})]-e^{-a^{2}}\sum _{k=1}^{n}I_{k}(a^{2}),$

for any non-negative integer $n$.

• For $\nu =1/2$, using the basic integral definition of generalized Marcum Q-function, we have

$Q_{1/2}(a,b)={\frac {1}{2}}\left[\mathrm {erfc} \left({\frac {b-a}{\sqrt {2}}}\right)+\mathrm {erfc} \left({\frac {b+a}{\sqrt {2}}}\right)\right].$

• For $\nu =3/2$, we have

$Q_{3/2}(a,b)=Q_{1/2}(a,b)+{\sqrt {\frac {2}{\pi }}}\,{\frac {\sinh(ab)}{a}}e^{-(a^{2}+b^{2})/2}.$

• For $\nu =5/2$ we have

$Q_{5/2}(a,b)=Q_{3/2}(a,b)+{\sqrt {\frac {2}{\pi }}}\,{\frac {ab\cosh(ab)-\sinh(ab)}{a^{3}}}e^{-(a^{2}+b^{2})/2}.$

Asymptotic forms

• Assuming $\nu$ to be fixed and $ab$ large, let $\zeta =a/b>0$, then the generalized Marcum-Q function has the following asymptotic form

$Q_{\nu }(a,b)\sim \sum _{n=0}^{\infty }\psi _{n},$

where $\psi _{n}$ is given by

$\psi _{n}={\frac {1}{2\zeta ^{\nu }{\sqrt {2\pi }}}}(-1)^{n}\left[A_{n}(\nu -1)-\zeta A_{n}(\nu )\right]\phi _{n}.$

The functions $\phi _{n}$ and $A_{n}$ are given by

$\phi _{n}=\left[{\frac {(b-a)^{2}}{2ab}}\right]^{n-{\frac {1}{2}}}\Gamma \left({\frac {1}{2}}-n,{\frac {(b-a)^{2}}{2}}\right),$

$A_{n}(\nu )={\frac {2^{-n}\Gamma ({\frac {1}{2}}+\nu +n)}{n!\Gamma ({\frac {1}{2}}+\nu -n)}}.$

The function $A_{n}(\nu )$ satisfies the recursion

$A_{n+1}(\nu )=-{\frac {(2n+1)^{2}-4\nu ^{2}}{8(n+1)}}A_{n}(\nu ),$

for $n\geq 0$ and $A_{0}(\nu )=1.$

• In the first term of the above asymptotic approximation, we have

$\phi _{0}={\frac {\sqrt {2\pi ab}}{b-a}}\mathrm {erfc} \left({\frac {b-a}{\sqrt {2}}}\right).$

Hence, assuming $b>a$, the first term asymptotic approximation of the generalized Marcum-Q function is

$Q_{\nu }(a,b)\sim \psi _{0}=\left({\frac {b}{a}}\right)^{\nu -{\frac {1}{2}}}Q(b-a),$

where $Q(\cdot )$ is the Gaussian Q-function. Here $Q_{\nu }(a,b)\sim 0.5$ as $a\uparrow b.$

For the case when $a>b$, we have

$Q_{\nu }(a,b)\sim 1-\psi _{0}=1-\left({\frac {b}{a}}\right)^{\nu -{\frac {1}{2}}}Q(a-b).$

Here too $Q_{\nu }(a,b)\sim 0.5$ as $a\downarrow b.$

Differentiation

• The partial derivative of $Q_{\nu }(a,b)$ with respect to $a$ and $b$ is given by

${\frac {\partial }{\partial a}}Q_{\nu }(a,b)=a\left[Q_{\nu +1}(a,b)-Q_{\nu }(a,b)\right]=a\left({\frac {b}{a}}\right)^{\nu }e^{-(a^{2}+b^{2})/2}I_{\nu }(ab),$

${\frac {\partial }{\partial b}}Q_{\nu }(a,b)=b\left[Q_{\nu -1}(a,b)-Q_{\nu }(a,b)\right]=-b\left({\frac {b}{a}}\right)^{\nu -1}e^{-(a^{2}+b^{2})/2}I_{\nu -1}(ab).$

We can relate the two partial derivatives as

${\frac {1}{a}}{\frac {\partial }{\partial a}}Q_{\nu }(a,b)+{\frac {1}{b}}{\frac {\partial }{\partial b}}Q_{\nu +1}(a,b)=0.$

• The n-th partial derivative of $Q_{\nu }(a,b)$ with respect to its arguments is given by

${\frac {\partial ^{n}}{\partial a^{n}}}Q_{\nu }(a,b)=n!(-a)^{n}\sum _{k=0}^{[n/2]}{\frac {(-2a^{2})^{-k}}{k!(n-2k)!}}\sum _{p=0}^{n-k}(-1)^{p}{\binom {n-k}{p}}Q_{\nu +p}(a,b),$

${\frac {\partial ^{n}}{\partial b^{n}}}Q_{\nu }(a,b)={\frac {n!a^{1-\nu }}{2^{n}b^{n-\nu +1}}}e^{-(a^{2}+b^{2})/2}\sum _{k=[n/2]}^{n}{\frac {(-2b^{2})^{k}}{(n-k)!(2k-n)!}}\sum _{p=0}^{k-1}{\binom {k-1}{p}}\left(-{\frac {a}{b}}\right)^{p}I_{\nu -p-1}(ab).$

Inequalities

• The generalized Marcum-Q function satisfies a Turán-type inequality

$Q_{\nu }^{2}(a,b)>{\frac {Q_{\nu -1}(a,b)+Q_{\nu +1}(a,b)}{2}}>Q_{\nu -1}(a,b)Q_{\nu +1}(a,b)$

for all $a\geq b>0$ and $\nu >1$.

Bounds

Based on monotonicity and log-concavity

Various upper and lower bounds of generalized Marcum-Q function can be obtained using monotonicity and log-concavity of the function $\nu \mapsto Q_{\nu }(a,b)$ and the fact that we have closed form expression for $Q_{\nu }(a,b)$ when $\nu$ is half-integer valued.

Let $\lfloor x\rfloor _{0.5}$ and $\lceil x\rceil _{0.5}$ denote the pair of half-integer rounding operators that map a real $x$ to its nearest left and right half-odd integer, respectively, according to the relations

$\lfloor x\rfloor _{0.5}=\lfloor x-0.5\rfloor +0.5$

$\lceil x\rceil _{0.5}=\lceil x+0.5\rceil -0.5$

where $\lfloor x\rfloor$ and $\lceil x\rceil$ denote the integer floor and ceiling functions.

• The monotonicity of the function $\nu \mapsto Q_{\nu }(a,b)$ for all $a\geq 0$ and $b>0$ gives us the following simple bound

$Q_{\lfloor \nu \rfloor _{0.5}}(a,b)<Q_{\nu }(a,b)<Q_{\lceil \nu \rceil _{0.5}}(a,b).$

However, the relative error of this bound does not tend to zero when $b\to \infty$. For integral values of $\nu =n$, this bound reduces to

$Q_{n-0.5}(a,b)<Q_{n}(a,b)<Q_{n+0.5}(a,b).$

A very good approximation of the generalized Marcum Q-function for integer valued $\nu =n$ is obtained by taking the arithmetic mean of the upper and lower bound

$Q_{n}(a,b)\approx {\frac {Q_{n-0.5}(a,b)+Q_{n+0.5}(a,b)}{2}}.$

• A tighter bound can be obtained by exploiting the log-concavity of $\nu \mapsto Q_{\nu }(a,b)$ on $[1,\infty )$ as

$Q_{\nu _{1}}(a,b)^{\nu _{2}-v}Q_{\nu _{2}}(a,b)^{v-\nu _{1}}<Q_{\nu }(a,b)<{\frac {Q_{\nu _{2}}(a,b)^{\nu _{2}-\nu +1}}{Q_{\nu _{2}+1}(a,b)^{\nu _{2}-\nu }}},$

where $\nu _{1}=\lfloor \nu \rfloor _{0.5}$ and $\nu _{2}=\lceil \nu \rceil _{0.5}$ for $\nu \geq 1.5$. The tightness of this bound improves as either $a$ or $\nu$ increases. The relative error of this bound converges to 0 as $b\to \infty$. For integral values of $\nu =n$, this bound reduces to

${\sqrt {Q_{n-0.5}(a,b)Q_{n+0.5}(a,b)}}<Q_{n}(a,b)<Q_{n+0.5}(a,b){\sqrt {\frac {Q_{n+0.5}(a,b)}{Q_{n+1.5}(a,b)}}}.$

Cauchy-Schwarz bound

Using the trigonometric integral representation for integer valued $\nu =n$, the following Cauchy-Schwarz bound can be obtained

$e^{-b^{2}/2}\leq Q_{n}(a,b)\leq \exp \left[-{\frac {1}{2}}(b^{2}+a^{2})\right]{\sqrt {I_{0}(2ab)}}{\sqrt {{\frac {2n-1}{2}}+{\frac {\zeta ^{2(1-n)}}{2(1-\zeta ^{2})}}}},\qquad \zeta <1,$

$1-Q_{n}(a,b)\leq \exp \left[-{\frac {1}{2}}(b^{2}+a^{2})\right]{\sqrt {I_{0}(2ab)}}{\sqrt {\frac {\zeta ^{2(1-n)}}{2(\zeta ^{2}-1)}}},\qquad \zeta >1,$

where $\zeta =a/b>0$.

Exponential-type bounds

For analytical purpose, it is often useful to have bounds in simple exponential form, even though they may not be the tightest bounds achievable. Letting $\zeta =a/b>0$, one such bound for integer valued $\nu =n$ is given as

$e^{-(b+a)^{2}/2}\leq Q_{n}(a,b)\leq e^{-(b-a)^{2}/2}+{\frac {\zeta ^{1-n}-1}{\pi (1-\zeta )}}\left[e^{-(b-a)^{2}/2}-e^{-(b+a)^{2}/2}\right],\qquad \zeta <1,$

$Q_{n}(a,b)\geq 1-{\frac {1}{2}}\left[e^{-(a-b)^{2}/2}-e^{-(a+b)^{2}/2}\right],\qquad \zeta >1.$

When $n=1$, the bound simplifies to give

$e^{-(b+a)^{2}/2}\leq Q_{1}(a,b)\leq e^{-(b-a)^{2}/2},\qquad \zeta <1,$

$1-{\frac {1}{2}}\left[e^{-(a-b)^{2}/2}-e^{-(a+b)^{2}/2}\right]\leq Q_{1}(a,b),\qquad \zeta >1.$

Another such bound obtained via Cauchy-Schwarz inequality is given as

$e^{-b^{2}/2}\leq Q_{n}(a,b)\leq {\frac {1}{2}}{\sqrt {{\frac {2n-1}{2}}+{\frac {\zeta ^{2(1-n)}}{2(1-\zeta ^{2})}}}}\left[e^{-(b-a)^{2}/2}+e^{-(b+a)^{2}/2}\right],\qquad \zeta <1$

$Q_{n}(a,b)\geq 1-{\frac {1}{2}}{\sqrt {\frac {\zeta ^{2(1-n)}}{2(\zeta ^{2}-1)}}}\left[e^{-(b-a)^{2}/2}+e^{-(b+a)^{2}/2}\right],\qquad \zeta >1.$

Chernoff-type bound

Chernoff-type bounds for the generalized Marcum Q-function, where $\nu =n$ is an integer, is given by

$(1-2\lambda )^{-n}\exp \left(-\lambda b^{2}+{\frac {\lambda na^{2}}{1-2\lambda }}\right)\geq \left\{{\begin{array}{lr}Q_{n}(a,b),&b^{2}>n(a^{2}+2)\\1-Q_{n}(a,b),&b^{2}<n(a^{2}+2)\end{array}}\right.$

where the Chernoff parameter $(0<\lambda <1/2)$ has optimum value $\lambda _{0}$ of

$\lambda _{0}={\frac {1}{2}}\left(1-{\frac {n}{b^{2}}}-{\frac {n}{b^{2}}}{\sqrt {1+{\frac {(ab)^{2}}{n}}}}\right).$

Semi-linear approximation

The first-order Marcum-Q function can be semi-linearly approximated by

${\begin{aligned}Q_{1}(a,b)={\begin{cases}1,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mathrm {if} ~b<c_{1}\\-\beta _{0}e^{-{\frac {1}{2}}\left(a^{2}+\left(\beta _{0}\right)^{2}\right)}I_{0}\left(a\beta _{0}\right)\left(b-\beta _{0}\right)+Q_{1}\left(a,\beta _{0}\right),~~~~~\mathrm {if} ~c_{1}\leq b\leq c_{2}\\0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mathrm {if} ~b>c_{2}\end{cases}}\end{aligned}}$

where

${\begin{aligned}\beta _{0}={\frac {a+{\sqrt {a^{2}+2}}}{2}},\end{aligned}}$

${\begin{aligned}c_{1}(a)=\max {\Bigg (}0,\beta _{0}+{\frac {Q_{1}\left(a,\beta _{0}\right)-1}{\beta _{0}e^{-{\frac {1}{2}}\left(a^{2}+\left(\beta _{0}\right)^{2}\right)}I_{0}\left(a\beta _{0}\right)}}{\Bigg )},\end{aligned}}$

and

${\begin{aligned}c_{2}(a)=\beta _{0}+{\frac {Q_{1}\left(a,\beta _{0}\right)}{\beta _{0}e^{-{\frac {1}{2}}\left(a^{2}+\left(\beta _{0}\right)^{2}\right)}I_{0}\left(a\beta _{0}\right)}}.\end{aligned}}$

Equivalent forms for efficient computation

It is convenient to re-express the Marcum Q-function as

$P_{N}(X,Y)=Q_{N}({\sqrt {2NX}},{\sqrt {2Y}}).$

The $P_{N}(X,Y)$ can be interpreted as the detection probability of $N$ incoherently integrated received signal samples of constant received signal-to-noise ratio, $X$, with a normalized detection threshold $Y$. In this equivalent form of Marcum Q-function, for given $a$ and $b$, we have $X=a^{2}/2N$ and $Y=b^{2}/2$. Many expressions exist that can represent $P_{N}(X,Y)$. However, the five most reliable, accurate, and efficient ones for numerical computation are given below. They are form one:

$P_{N}(X,Y)=\sum _{k=0}^{\infty }e^{-NX}{\frac {(NX)^{k}}{k!}}\sum _{m=0}^{N-1+k}e^{-Y}{\frac {Y^{m}}{m!}},$

form two:

$P_{N}(X,Y)=\sum _{m=0}^{N-1}e^{-Y}{\frac {Y^{m}}{m!}}+\sum _{m=N}^{\infty }e^{-Y}{\frac {Y^{m}}{m!}}\left(1-\sum _{k=0}^{m-N}e^{-NX}{\frac {(NX)^{k}}{k!}}\right),$

form three:

$1-P_{N}(X,Y)=\sum _{m=N}^{\infty }e^{-Y}{\frac {Y^{m}}{m!}}\sum _{k=0}^{m-N}e^{-NX}{\frac {(NX)^{k}}{k!}},$

form four:

$1-P_{N}(X,Y)=\sum _{k=0}^{\infty }e^{-NX}{\frac {(NX)^{k}}{k!}}\left(1-\sum _{m=0}^{N-1+k}e^{-Y}{\frac {Y^{m}}{m!}}\right),$

and form five:

$1-P_{N}(X,Y)=e^{-(NX+Y)}\sum _{r=N}^{\infty }\left({\frac {Y}{NX}}\right)^{r/2}I_{r}(2{\sqrt {NXY}}).$

Among these five form, the second form is the most robust.

Applications

The generalized Marcum Q-function can be used to represent the cumulative distribution function (cdf) of many random variables:

• If $X\sim \mathrm {Exp} (\lambda )$ is a exponential distribution with rate parameter $\lambda$, then its cdf is given by $F_{X}(x)=1-Q_{1}\left(0,{\sqrt {2\lambda x}}\right)$

• If $X\sim \mathrm {Erlang} (k,\lambda )$ is a Erlang distribution with shape parameter $k$ and rate parameter $\lambda$, then its cdf is given by $F_{X}(x)=1-Q_{k}\left(0,{\sqrt {2\lambda x}}\right)$

• If $X\sim \chi _{k}^{2}$ is a chi-squared distribution with $k$ degrees of freedom, then its cdf is given by $F_{X}(x)=1-Q_{k/2}(0,{\sqrt {x}})$

• If $X\sim \mathrm {Gamma} (\alpha ,\beta )$ is a gamma distribution with shape parameter $\alpha$ and rate parameter $\beta$, then its cdf is given by $F_{X}(x)=1-Q_{\alpha }(0,{\sqrt {2\beta x}})$

• If $X\sim \mathrm {Weibull} (k,\lambda )$ is a Weibull distribution with shape parameters $k$ and scale parameter $\lambda$, then its cdf is given by $F_{X}(x)=1-Q_{1}\left(0,{\sqrt {2}}\left({\frac {x}{\lambda }}\right)^{\frac {k}{2}}\right)$

• If $X\sim \mathrm {GG} (a,d,p)$ is a generalized gamma distribution with parameters $a,d,p$, then its cdf is given by $F_{X}(x)=1-Q_{\frac {d}{p}}\left(0,{\sqrt {2}}\left({\frac {x}{a}}\right)^{\frac {p}{2}}\right)$

• If $X\sim \chi _{k}^{2}(\lambda )$ is a non-central chi-squared distribution with non-centrality parameter $\lambda$ and $k$ degrees of freedom, then its cdf is given by $F_{X}(x)=1-Q_{k/2}({\sqrt {\lambda }},{\sqrt {x}})$

• If $X\sim \mathrm {Rayleigh} (\sigma )$ is a Rayleigh distribution with parameter $\sigma$, then its cdf is given by $F_{X}(x)=1-Q_{1}\left(0,{\frac {x}{\sigma }}\right)$

• If $X\sim \mathrm {Maxwell} (\sigma )$ is a Maxwell–Boltzmann distribution with parameter $\sigma$, then its cdf is given by $F_{X}(x)=1-Q_{3/2}\left(0,{\frac {x}{\sigma }}\right)$

• If $X\sim \chi _{k}$ is a chi distribution with $k$ degrees of freedom, then its cdf is given by $F_{X}(x)=1-Q_{k/2}(0,x)$

• If $X\sim \mathrm {Nakagami} (m,\Omega )$ is a Nakagami distribution with $m$ as shape parameter and $\Omega$ as spread parameter, then its cdf is given by $F_{X}(x)=1-Q_{m}\left(0,{\sqrt {\frac {2m}{\Omega }}}x\right)$

• If $X\sim \mathrm {Rice} (\nu ,\sigma )$ is a Rice distribution with parameters $\nu$ and $\sigma$, then its cdf is given by $F_{X}(x)=1-Q_{1}\left({\frac {\nu }{\sigma }},{\frac {x}{\sigma }}\right)$

• If $X\sim \chi _{k}(\lambda )$ is a non-central chi distribution with non-centrality parameter $\lambda$ and $k$ degrees of freedom, then its cdf is given by $F_{X}(x)=1-Q_{k/2}(\lambda ,x)$

References

• Marcum, J. I. (1950) "Table of Q Functions". U.S. Air Force RAND Research Memorandum M-339. Santa Monica, CA: Rand Corporation, Jan. 1, 1950.
• Nuttall, Albert H. (1975): Some Integrals Involving the Q_M Function, IEEE Transactions on Information Theory, 21(1), 95–96, ISSN 0018-9448
• Shnidman, David A. (1989): The Calculation of the Probability of Detection and the Generalized Marcum Q-Function, IEEE Transactions on Information Theory, 35(2), 389-400.
• Weisstein, Eric W. Marcum Q-Function. From MathWorld—A Wolfram Web Resource. [1]