Marcum Q-function

Short description: Function in statistics

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

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

where $b\ge 0$ 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 $\nu =1$ by, and hence named after, 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

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

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

$Q_n(a,b)=\{\begin{array}{lr}{H}_n(a,b)& a<b,\\ \frac{1}{2}+H_n(a,a)& a=b,\\ 1+H_n(a,b)& a>b,\end{array}$

where

$H_n(a,b)=\frac{{\zeta }^{1-n}}{2\pi }\exp (-\frac{a^2+b^2}{2})\underset{0}{\overset{2\pi }{\int }}\frac{\cos (n-1)\theta -\zeta \cos n\theta }{1-2\zeta \cos \theta +{\zeta }^2}\exp (ab\cos \theta )\mathrm{d}\theta ,$

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

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

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

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

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

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

Series representation

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

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

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

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

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

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

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

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

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

where $\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(\cdot )$ 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:

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

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 $Q_{\mathrm{\infty }}(a,b)=1$ and $Q_{-\mathrm{\infty }}(a,b)=0$ for $n=\mathrm{\infty }$, 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

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

• 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

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

where $|t|<1.$

Symmetry relation

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

$Q_n(a,b)+Q_n(b,a)=1+e^{-(a^2+b^2)/2}[I_0(ab)+\sum _{k=1}^{n-1}\frac{a^{2k}+b^{2k}}{(ab{)}^k}{I}_k(ab)].$

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

• $Q_{\nu }(0,0)=1,$
• $Q_{\nu }(a,0)=1,$
• $Q_{\nu }(a,+\mathrm{\infty })=0,$
• $Q_{\nu }(0,b)=\frac{\mathrm{\Gamma }(\nu ,b^2/2)}{\mathrm{\Gamma }(\nu )},$
• $Q_{\nu }(+\mathrm{\infty },b)=1,$
• $Q_{\mathrm{\infty }}(a,b)=1,$
• 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

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

• 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

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

where ${\psi }_n$ is given by

${\psi }_n=\frac{1}{2{\zeta }^{\nu }\sqrt{2\pi }}(-1{)}^n[A_n(\nu -1)-\zeta A_n(\nu )]{\varphi }_n.$

The functions ${\varphi }_n$ and $A_n$ are given by

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

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

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 $n\ge 0$ and $A_0(\nu )=1.$

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

${\varphi }_0=\frac{\sqrt{2\pi ab}}{b-a}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}\left(\frac{b-a}{\sqrt{2}}\right).$

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

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

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

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

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

$\frac{{\partial }^n}{\partial b^n}{Q}_{\nu }(a,b)=\frac{n!a^{1-\nu }}{2^n{b}^{n-\nu +1}}{e}^{-(a^2+b^2)/2}\sum _{k=[n/2]}^n\frac{(-2b^2{)}^k}{(n-k)!(2k-n)!}\sum _{p=0}^{k-1}(\genfrac{}{}{0ex}{}{k-1}{p}){(-\frac{a}{b})}^p{I}_{\nu -p-1}(ab).$

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 $a\ge b>0$ 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 $a\ge 0$ 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 $b\to \mathrm{\infty }$. 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 $[1,\mathrm{\infty })$ 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 $\nu \ge 1.5$. The tightness of this bound improves as either $a$ or $\nu$ increases. The relative error of this bound converges to 0 as $b\to \mathrm{\infty }$. 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

$e^{-b^2/2}\le Q_n(a,b)\le \exp [-\frac{1}{2}(b^2+a^2)]\sqrt{I_0(2ab)}\sqrt{\frac{2n-1}{2}+\frac{{\zeta }^{2(1-n)}}{2(1-{\zeta }^2)}},\zeta <1,$

$1-Q_n(a,b)\le \exp [-\frac{1}{2}(b^2+a^2)]\sqrt{I_0(2ab)}\sqrt{\frac{{\zeta }^{2(1-n)}}{2({\zeta }^2-1)}},\zeta >1,$

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

$e^{-(b+a{)}^2/2}\le Q_n(a,b)\le e^{-(b-a{)}^2/2}+\frac{{\zeta }^{1-n}-1}{\pi (1-\zeta )}[e^{-(b-a{)}^2/2}-e^{-(b+a{)}^2/2}],\zeta <1,$

$Q_n(a,b)\ge 1-\frac{1}{2}[e^{-(a-b{)}^2/2}-e^{-(a+b{)}^2/2}],\zeta >1.$

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

$e^{-(b+a{)}^2/2}\le Q_1(a,b)\le e^{-(b-a{)}^2/2},\zeta <1,$

$1-\frac{1}{2}[e^{-(a-b{)}^2/2}-e^{-(a+b{)}^2/2}]\le Q_1(a,b),\zeta >1.$

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

$e^{-b^2/2}\le Q_n(a,b)\le \frac{1}{2}\sqrt{\frac{2n-1}{2}+\frac{{\zeta }^{2(1-n)}}{2(1-{\zeta }^2)}}[e^{-(b-a{)}^2/2}+e^{-(b+a{)}^2/2}],\zeta <1$

$Q_n(a,b)\ge 1-\frac{1}{2}\sqrt{\frac{{\zeta }^{2(1-n)}}{2({\zeta }^2-1)}}[e^{-(b-a{)}^2/2}+e^{-(b+a{)}^2/2}],\zeta >1.$

Chernoff-type bound

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

$(1-2\lambda {)}^{-n}\exp (-\lambda b^2+\frac{\lambda n{a}^2}{1-2\lambda })\ge \{\begin{array}{lr}{Q}_n(a,b),& b^2>n(a^2+2)\\ 1-Q_n(a,b),& b^2<n(a^2+2)\end{array}$

where the Chernoff parameter $(0<\lambda <1/2)$ has optimum value ${\lambda }_0$ of

${\lambda }_0=\frac{1}{2}(1-\frac{n}{b^2}-\frac{n}{b^2}\sqrt{1+\frac{(ab{)}^2}{n}}).$

Semi-linear approximation

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

$\begin{array}{r}{Q}_1(a,b)=\{\begin{array}{l}1,\mathrm{i}\mathrm{f}b<c_1\\ -{\beta }_0{e}^{-\frac{1}{2}(a^2+{\left({\beta }_0\right)}^2)}{I}_0\left(a{\beta }_0\right)(b-{\beta }_0)+Q_1(a,{\beta }_0),\mathrm{i}\mathrm{f}{c}_1\le b\le c_2\\ 0,\mathrm{i}\mathrm{f}b>c_2\end{array}\end{array}$

where

$\begin{array}{r}{\beta }_0=\frac{a+\sqrt{a^2+2}}{2},\end{array}$

$\begin{array}{r}{c}_1(a)=\max (0,{\beta }_0+\frac{Q_1(a,{\beta }_0)-1}{{\beta }_0{e}^{-\frac{1}{2}(a^2+{\left({\beta }_0\right)}^2)}{I}_0\left(a{\beta }_0\right)}),\end{array}$

and

$\begin{array}{r}{c}_2(a)={\beta }_0+\frac{Q_1(a,{\beta }_0)}{{\beta }_0{e}^{-\frac{1}{2}(a^2+{\left({\beta }_0\right)}^2)}{I}_0\left(a{\beta }_0\right)}.\end{array}$

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:

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

form two:

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

form three:

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

form four:

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

and form five:

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

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 $X\sim \mathrm{E}\mathrm{x}\mathrm{p}(\lambda )$ is an exponential distribution with rate parameter $\lambda$, then its cdf is given by $F_X(x)=1-Q_1(0,\sqrt{2\lambda x})$
• If $X\sim \mathrm{E}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{g}(k,\lambda )$ is a Erlang distribution with shape parameter $k$ and rate parameter $\lambda$, then its cdf is given by $F_X(x)=1-Q_k(0,\sqrt{2\lambda x})$
• 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 $X\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(\alpha ,\beta )$ 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 $X\sim \mathrm{W}\mathrm{e}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{l}(k,\lambda )$ is a Weibull distribution with shape parameters $k$ and scale parameter $\lambda$, then its cdf is given by $F_X(x)=1-Q_1(0,\sqrt{2}{\left(\frac{x}{\lambda }\right)}^{\frac{k}{2}})$
• If $X\sim \mathrm{G}\mathrm{G}(a,d,p)$ is a generalized gamma distribution with parameters $a,d,p$, then its cdf is given by $F_X(x)=1-Q_{\frac{d}{p}}(0,\sqrt{2}{\left(\frac{x}{a}\right)}^{\frac{p}{2}})$
• 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 $X\sim \mathrm{R}\mathrm{a}\mathrm{y}\mathrm{l}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}(\sigma )$ is a Rayleigh distribution with parameter $\sigma$, then its cdf is given by $F_X(x)=1-Q_1(0,\frac{x}{\sigma })$
• If $X\sim \mathrm{M}\mathrm{a}\mathrm{x}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}(\sigma )$ is a Maxwell–Boltzmann distribution with parameter $\sigma$, then its cdf is given by $F_X(x)=1-Q_{3/2}(0,\frac{x}{\sigma })$
• 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 $X\sim \mathrm{N}\mathrm{a}\mathrm{k}\mathrm{a}\mathrm{g}\mathrm{a}\mathrm{m}\mathrm{i}(m,\mathrm{\Omega })$ is a Nakagami distribution with $m$ as shape parameter and $\mathrm{\Omega }$ as spread parameter, then its cdf is given by $F_X(x)=1-Q_m(0,\sqrt{\frac{2m}{\mathrm{\Omega }}}x)$
• If $X\sim \mathrm{R}\mathrm{i}\mathrm{c}\mathrm{e}(\nu ,\sigma )$ is a Rice distribution with parameters $\nu$ and $\sigma$, then its cdf is given by $F_X(x)=1-Q_1(\frac{\nu }{\sigma },\frac{x}{\sigma })$
• 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]