Marcum Q-function

In statistics, the generalized Marcum Q-function of order [math]\displaystyle{ \nu }[/math] is defined as

[math]\displaystyle{ 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 }[/math]

where [math]\displaystyle{ b \geq 0 }[/math] and [math]\displaystyle{ a, \nu \gt 0 }[/math] and [math]\displaystyle{ I_{\nu-1} }[/math] is the modified Bessel function of first kind of order [math]\displaystyle{ \nu-1 }[/math]. If [math]\displaystyle{ b \gt 0 }[/math], the integral converges for any [math]\displaystyle{ \nu }[/math]. 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 [math]\displaystyle{ \nu = 1 }[/math], and hence named after, by Jess Marcum for pulsed radars.

Properties

Finite integral representation

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

[math]\displaystyle{ 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. }[/math]

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 integral value of [math]\displaystyle{ \nu = n }[/math], such a representation is given by the trigonometric integral

[math]\displaystyle{ Q_n(a,b) = \left\{ \begin{array}{lr} H_n(a,b) & a \lt b, \\ \frac{1}{2} + H_n(a,a) & a=b, \\ 1 + H_n(a,b) & a \gt b, \end{array} \right. }[/math]

where

[math]\displaystyle{ 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, }[/math]

and the ratio [math]\displaystyle{ \zeta = a/b }[/math] is a constant.

For any real [math]\displaystyle{ \nu \gt 0 }[/math], such finite trigonometric integral is given by

[math]\displaystyle{ Q_\nu(a,b) = \left\{ \begin{array}{lr} H_\nu(a,b) + C_\nu(a,b) & a \lt b, \\ \frac{1}{2} + H_\nu(a,a) + C_\nu(a,b) & a=b, \\ 1 + H_\nu(a,b) + C_\nu(a,b) & a \gt b, \end{array} \right. }[/math]

where [math]\displaystyle{ H_n(a,b) }[/math] is as defined before, [math]\displaystyle{ \zeta = a/b }[/math], and the additional correction term is given by

[math]\displaystyle{ 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. }[/math]

For integer values of [math]\displaystyle{ \nu }[/math], the correction term [math]\displaystyle{ C_\nu(a,b) }[/math] tend to vanish.

Monotonicity and log-concavity

• The generalized Marcum Q-function [math]\displaystyle{ Q_\nu(a,b) }[/math] is strictly increasing in [math]\displaystyle{ \nu }[/math] and [math]\displaystyle{ a }[/math] for all [math]\displaystyle{ a \geq 0 }[/math] and [math]\displaystyle{ b, \nu \gt 0 }[/math], and is strictly decreasing in [math]\displaystyle{ b }[/math] for all [math]\displaystyle{ a, b \geq 0 }[/math] and [math]\displaystyle{ \nu\gt 0. }[/math]

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

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

• The function [math]\displaystyle{ a \mapsto 1 - Q_\nu(a,b) }[/math] is log-concave on [math]\displaystyle{ [0,\infty) }[/math] for all [math]\displaystyle{ b, \nu \gt 0. }[/math]

Series representation

• The generalized Marcum Q function of order [math]\displaystyle{ \nu \gt 0 }[/math] can be represented using incomplete Gamma function as

[math]\displaystyle{ 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, }[/math]

where [math]\displaystyle{ \gamma(s,x) }[/math] is the lower incomplete Gamma function. This is usually called the canonical representation of the [math]\displaystyle{ \nu }[/math]-th order generalized Marcum Q-function.

• The generalized Marcum Q function of order [math]\displaystyle{ \nu \gt 0 }[/math] can also be represented using generalized Laguerre polynomials as

[math]\displaystyle{ 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}, }[/math]

where [math]\displaystyle{ L_k^{(\alpha)}(\cdot) }[/math] is the generalized Laguerre polynomial of degree [math]\displaystyle{ k }[/math] and of order [math]\displaystyle{ \alpha }[/math].

• The generalized Marcum Q-function of order [math]\displaystyle{ \nu \gt 0 }[/math] can also be represented as Neumann series expansions

[math]\displaystyle{ Q_\nu (a,b) = e^{-(a^2 + b^2)/2} \sum_{\alpha=1-\nu}^\infty \left( \frac{a}{b}\right)^\alpha I_{-\alpha}(ab), }[/math]

[math]\displaystyle{ 1 - Q_\nu(a,b) = e^{-(a^2 + b^2)/2} \sum_{\alpha=\nu}^\infty \left( \frac{b}{a}\right)^\alpha I_{\alpha}(ab), }[/math]

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

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

[math]\displaystyle{ 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), }[/math]

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

[math]\displaystyle{ 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], }[/math]

where [math]\displaystyle{ n }[/math] is non-negative integer, we can exactly represent the generalized Marcum Q-function with half-integer parameter. More precisely, we have

[math]\displaystyle{ 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], }[/math]

for non-negative integers [math]\displaystyle{ n }[/math], where [math]\displaystyle{ Q(\cdot) }[/math] 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:

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

where [math]\displaystyle{ g_0(z) = z^{-1} }[/math], [math]\displaystyle{ g_1(z) = -z^{-2} }[/math], and [math]\displaystyle{ g_{n-1}(z) - g_{n+1}(z) = (2n+1) z^{-1} g_n(z) }[/math] for any integer value of [math]\displaystyle{ n }[/math].

Recurrence relation and generating function

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

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

• The above formula is easily generalized as

[math]\displaystyle{ 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), }[/math]

[math]\displaystyle{ 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), }[/math]

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

• The related three-term recurrence relation is given by

[math]\displaystyle{ Q_{\nu+1}(a,b) - (1+c_\nu(a,b))Q_\nu(a,b) + c_\nu(a,b) Q_{\nu-1}(a,b) = 0, }[/math]

where

[math]\displaystyle{ c_\nu(a,b) = \left(\frac{b}{a}\right) \frac{I_\nu(ab)}{I_{\nu+1}(ab)}. }[/math]

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

[math]\displaystyle{ \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). }[/math]

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

[math]\displaystyle{ Q_{\nu+1}(a,b) = Q_\nu(a,b) + \frac{1}{a} \frac{\partial}{\partial a} Q_\nu(a,b), }[/math]

[math]\displaystyle{ Q_{\nu-1}(a,b) = Q_\nu(a,b) + \frac{1}{b} \frac{\partial}{\partial b} Q_\nu(a,b). }[/math]

• The ordinary generating function of [math]\displaystyle{ Q_\nu(a,b) }[/math] for integral [math]\displaystyle{ \nu }[/math] is

[math]\displaystyle{ \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}, }[/math]

where [math]\displaystyle{ |t|\lt 1. }[/math]

Symmetry relation

• Using the two Neumann series representations, we can obtain the following symmetry relation for positive integral [math]\displaystyle{ \nu = n }[/math]

[math]\displaystyle{ 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]. }[/math]

In particular, for [math]\displaystyle{ n = 1 }[/math] we have

[math]\displaystyle{ Q_1(a,b) + Q_1(b,a) = 1 + e^{-(a^2+b^2)/2} I_0(ab). }[/math]

Special values

Some specific values of Marcum-Q function are

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

[math]\displaystyle{ Q_1(a,a) = \frac{1}{2}[1 + e^{-a^2}I_0(a^2)], }[/math]

which when combined with the recursive formula gives

[math]\displaystyle{ 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), }[/math]

[math]\displaystyle{ 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), }[/math]

for any non-negative integer [math]\displaystyle{ n }[/math].

• For [math]\displaystyle{ \nu = 1/2 }[/math], using the basic integral definition of generalized Marcum Q-function, we have

[math]\displaystyle{ 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]. }[/math]

• For [math]\displaystyle{ \nu=3/2 }[/math], we have

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

• For [math]\displaystyle{ \nu = 5/2 }[/math] we have

[math]\displaystyle{ 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}. }[/math]

Asymptotic forms

• Assuming [math]\displaystyle{ \nu }[/math] to be fixed and [math]\displaystyle{ ab }[/math] large, let [math]\displaystyle{ \zeta = a/b \gt 0 }[/math], then the generalized Marcum-Q function has the following asymptotic form

[math]\displaystyle{ Q_\nu(a,b) \sim \sum_{n=0}^\infty \psi_n, }[/math]

where [math]\displaystyle{ \psi_n }[/math] is given by

[math]\displaystyle{ \psi_n = \frac{1}{2\zeta^\nu \sqrt{2\pi}} (-1)^n \left[ A_n(\nu-1) - \zeta A_n(\nu) \right] \phi_n. }[/math]

The functions [math]\displaystyle{ \phi_n }[/math] and [math]\displaystyle{ A_n }[/math] are given by

[math]\displaystyle{ \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), }[/math]

[math]\displaystyle{ A_n(\nu) = \frac{2^{-n}\Gamma(\frac{1}{2}+\nu+n)}{n!\Gamma(\frac{1}{2}+\nu-n)}. }[/math]

The function [math]\displaystyle{ A_n(\nu) }[/math] satisfies the recursion

[math]\displaystyle{ A_{n+1}(\nu) = - \frac{(2n+1)^2 - 4\nu^2}{8(n+1)}A_n(\nu), }[/math]

for [math]\displaystyle{ n \geq 0 }[/math] and [math]\displaystyle{ A_0(\nu)=1. }[/math]

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

[math]\displaystyle{ \phi_0 = \frac{\sqrt{2 \pi ab}}{b-a} \mathrm{erfc}\left(\frac{b-a}{\sqrt{2}}\right). }[/math]

Hence, assuming [math]\displaystyle{ b\gt a }[/math], the first term asymptotic approximation of the generalized Marcum-Q function is

[math]\displaystyle{ Q_\nu(a,b) \sim \psi_0 = \left(\frac{b}{a}\right)^{\nu-\frac{1}{2}} Q(b-a), }[/math]

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

For the case when [math]\displaystyle{ a \gt b }[/math], we have

[math]\displaystyle{ Q_\nu(a,b) \sim 1-\psi_0 = 1-\left(\frac{b}{a}\right)^{\nu-\frac{1}{2}} Q(a-b). }[/math]

Here too [math]\displaystyle{ Q_\nu(a,b) \sim 0.5 }[/math] as [math]\displaystyle{ a \downarrow b. }[/math]

Differentiation

• The partial derivative of [math]\displaystyle{ Q_\nu(a,b) }[/math] with respect to [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] is given by

[math]\displaystyle{ \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), }[/math]

[math]\displaystyle{ \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). }[/math]

We can relate the two partial derivatives as

[math]\displaystyle{ \frac{1}{a}\frac{\partial}{\partial a} Q_\nu(a,b) + \frac{1}{b} \frac{\partial}{\partial b} Q_{\nu+1}(a,b) = 0. }[/math]

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

[math]\displaystyle{ \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), }[/math]

[math]\displaystyle{ \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). }[/math]

Inequalities

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

[math]\displaystyle{ Q^2_\nu(a,b) \gt \frac{Q_{\nu-1}(a,b) + Q_{\nu+1}(a,b)}{2} \gt Q_{\nu-1}(a,b) Q_{\nu+1}(a,b) }[/math]

for all [math]\displaystyle{ a \geq b \gt 0 }[/math] and [math]\displaystyle{ \nu \gt 1 }[/math].

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 [math]\displaystyle{ \nu \mapsto Q_\nu(a,b) }[/math] and the fact that we have closed form expression for [math]\displaystyle{ Q_\nu(a,b) }[/math] when [math]\displaystyle{ \nu }[/math] is half-integer valued.

Let [math]\displaystyle{ \lfloor x \rfloor_{0.5} }[/math] and [math]\displaystyle{ \lceil x \rceil_{0.5} }[/math] denote the pair of half-integer rounding operators that map a real [math]\displaystyle{ x }[/math] to its nearest left and right half-odd integer, respectively, according to the relations

[math]\displaystyle{ \lfloor x \rfloor_{0.5} = \lfloor x - 0.5 \rfloor + 0.5 }[/math]

[math]\displaystyle{ \lceil x \rceil_{0.5} = \lceil x + 0.5 \rceil - 0.5 }[/math]

where [math]\displaystyle{ \lfloor x \rfloor }[/math] and [math]\displaystyle{ \lceil x \rceil }[/math] denote the integer floor and ceiling functions.

• The monotonicity of the function [math]\displaystyle{ \nu \mapsto Q_\nu(a,b) }[/math] for all [math]\displaystyle{ a \geq 0 }[/math] and [math]\displaystyle{ b \gt 0 }[/math] gives us the following simple bound

[math]\displaystyle{ Q_{\lfloor\nu\rfloor_{0.5}}(a,b) \lt Q_\nu(a,b) \lt Q_{\lceil\nu\rceil_{0.5}}(a,b). }[/math]

However, the relative error of this bound does not tend to zero when [math]\displaystyle{ b \to \infty }[/math]. For integral values of [math]\displaystyle{ \nu = n }[/math], this bound reduces to

[math]\displaystyle{ Q_{n-0.5}(a,b) \lt Q_n(a,b) \lt Q_{n+0.5}(a,b). }[/math]

A very good approximation of the generalized Marcum Q-function for integer valued [math]\displaystyle{ \nu = n }[/math] is obtained by taking the arithmetic mean of the upper and lower bound

[math]\displaystyle{ Q_n(a,b) \approx \frac{Q_{n-0.5}(a,b) + Q_{n+0.5}(a,b)}{2}. }[/math]

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

[math]\displaystyle{ Q_{\nu_1}(a,b)^{\nu_2 - v} Q_{\nu_2}(a,b)^{v - \nu_1} \lt Q_\nu(a,b) \lt \frac{Q_{\nu_2}(a,b)^{\nu_2 - \nu + 1}}{Q_{\nu_2 + 1}(a,b)^{\nu_2 - \nu}}, }[/math]

where [math]\displaystyle{ \nu_1 = \lfloor\nu\rfloor_{0.5} }[/math] and [math]\displaystyle{ \nu_2 = \lceil\nu\rceil_{0.5} }[/math] for [math]\displaystyle{ \nu \geq 1.5 }[/math]. The tightness of this bound improves as either [math]\displaystyle{ a }[/math] or [math]\displaystyle{ \nu }[/math] increases. The relative error of this bound converges to 0 as [math]\displaystyle{ b \to \infty }[/math]. For integral values of [math]\displaystyle{ \nu = n }[/math], this bound reduces to

[math]\displaystyle{ \sqrt{Q_{n - 0.5}(a,b) Q_{n + 0.5}(a,b)} \lt Q_n(a,b) \lt Q_{n + 0.5}(a,b) \sqrt{\frac{Q_{n + 0.5}(a,b)}{Q_{n + 1.5}(a,b)}}. }[/math]

Cauchy-Schwarz bound

Using the trigonometric integral representation for integer valued [math]\displaystyle{ \nu=n }[/math], the following Cauchy-Schwarz bound can be obtained

[math]\displaystyle{ 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 \lt 1, }[/math]

[math]\displaystyle{ 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 \gt 1, }[/math]

where [math]\displaystyle{ \zeta = a/b \gt 0 }[/math].

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 [math]\displaystyle{ \zeta = a/b \gt 0 }[/math], one such bound for integer valued [math]\displaystyle{ \nu = n }[/math] is given as

[math]\displaystyle{ 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 \lt 1, }[/math]

[math]\displaystyle{ Q_n(a,b) \geq 1 - \frac{1}{2}\left[e^{-(a-b)^2/2} - e^{-(a+b)^2/2} \right], \qquad \zeta \gt 1. }[/math]

When [math]\displaystyle{ n=1 }[/math], the bound simplifies to give

[math]\displaystyle{ e^{-(b+a)^2/2} \leq Q_1(a,b) \leq e^{-(b-a)^2/2}, \qquad \zeta \lt 1, }[/math]

[math]\displaystyle{ 1 - \frac{1}{2}\left[e^{-(a-b)^2/2} - e^{-(a+b)^2/2} \right] \leq Q_1(a,b), \qquad \zeta \gt 1. }[/math]

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

[math]\displaystyle{ 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 \lt 1 }[/math]

[math]\displaystyle{ 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 \gt 1. }[/math]

Chernoff-type bound

Chernoff-type bounds for the generalized Marcum Q-function, where [math]\displaystyle{ \nu = n }[/math] is an integer, is given by

[math]\displaystyle{ (1-2\lambda)^{-n} \exp \left(-\lambda b^2 + \frac{\lambda n a^2}{1 - 2\lambda} \right) \geq \left\{ \begin{array}{lr} Q_n(a,b), & b^2 \gt n(a^2+2) \\ 1 - Q_n(a,b), & b^2 \lt n(a^2+2) \end{array} \right. }[/math]

where the Chernoff parameter [math]\displaystyle{ (0 \lt \lambda \lt 1/2) }[/math] has optimum value [math]\displaystyle{ \lambda_0 }[/math] of

[math]\displaystyle{ \lambda_0 = \frac{1}{2}\left(1 - \frac{n}{b^2} - \frac{n}{b^2} \sqrt{1 + \frac{(ab)^2}{n}}\right). }[/math]

Semi-linear approximation

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

[math]\displaystyle{ \begin{align} Q_1(a, b)= \begin{cases} 1, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mathrm{if}~ b \lt 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\gt c_2 \end{cases} \end{align} }[/math]

where

[math]\displaystyle{ \begin{align} \beta_0 = \frac{a+\sqrt{a^2+2}}{2}, \end{align} }[/math]

[math]\displaystyle{ \begin{align} 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{align} }[/math]

and

[math]\displaystyle{ \begin{align} 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{align} }[/math]

Equivalent forms for efficient computation

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

[math]\displaystyle{ P_N(X,Y) = Q_N(\sqrt{2NX},\sqrt{2Y}). }[/math]

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

[math]\displaystyle{ 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!}, }[/math]

form two:

[math]\displaystyle{ 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), }[/math]

form three:

[math]\displaystyle{ 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!}, }[/math]

form four:

[math]\displaystyle{ 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), }[/math]

and form five:

[math]\displaystyle{ 1 - P_N(X,Y) = e^{-(NX+Y)} \sum_{r=N}^\infty \left(\frac{Y}{NX}\right)^{r/2} I_r(2\sqrt{NXY}). }[/math]

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 [math]\displaystyle{ X \sim \mathrm{Exp}(\lambda) }[/math] is a exponential distribution with rate parameter [math]\displaystyle{ \lambda }[/math], then its cdf is given by [math]\displaystyle{ F_X(x) = 1 - Q_1\left(0,\sqrt{2 \lambda x}\right) }[/math]

• If [math]\displaystyle{ X \sim \mathrm{Erlang}(k,\lambda) }[/math] is a Erlang distribution with shape parameter [math]\displaystyle{ k }[/math] and rate parameter [math]\displaystyle{ \lambda }[/math], then its cdf is given by [math]\displaystyle{ F_X(x) = 1 - Q_k\left(0,\sqrt{2 \lambda x}\right) }[/math]

• If [math]\displaystyle{ X \sim \chi^2_k }[/math] is a chi-squared distribution with [math]\displaystyle{ k }[/math] degrees of freedom, then its cdf is given by [math]\displaystyle{ F_X(x) = 1 - Q_{k/2}(0,\sqrt{x}) }[/math]

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

• If [math]\displaystyle{ X \sim \mathrm{Weibull}(k,\lambda) }[/math] is a Weibull distribution with shape parameters [math]\displaystyle{ k }[/math] and scale parameter [math]\displaystyle{ \lambda }[/math], then its cdf is given by [math]\displaystyle{ F_X(x) = 1 - Q_1 \left( 0, \sqrt{2} \left(\frac{x}{\lambda}\right)^{\frac{k}{2}} \right) }[/math]

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

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

• If [math]\displaystyle{ X \sim \mathrm{Rayleigh}(\sigma) }[/math] is a Rayleigh distribution with parameter [math]\displaystyle{ \sigma }[/math], then its cdf is given by [math]\displaystyle{ F_X(x) = 1 - Q_1\left(0,\frac{x}{\sigma}\right) }[/math]

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

• If [math]\displaystyle{ X \sim \chi_k }[/math] is a chi distribution with [math]\displaystyle{ k }[/math] degrees of freedom, then its cdf is given by [math]\displaystyle{ F_X(x) = 1 - Q_{k/2}(0,x) }[/math]

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

• If [math]\displaystyle{ X \sim \mathrm{Rice}(\nu,\sigma) }[/math] is a Rice distribution with parameters [math]\displaystyle{ \nu }[/math] and [math]\displaystyle{ \sigma }[/math], then its cdf is given by [math]\displaystyle{ F_X(x) = 1 - Q_1\left(\frac{\nu}{\sigma},\frac{x}{\sigma}\right) }[/math]

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

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]