{{Short description|Function in statistics}} In statistics, the generalized '''Marcum Q-function''' of order <math>\nu</math> is defined as
: <math>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>b \geq 0</math> and <math>a, \nu > 0</math> and <math>I_{\nu-1}</math> is the modified Bessel function of first kind of order <math>\nu-1</math>. If <math>b > 0</math>, the integral converges for any <math>\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>\nu = 1</math> by, and hence named after, Jess Marcum for pulsed radars.<ref name="Marcum">J.I. Marcum (1960). A statistical theory of target detection by pulsed radar: mathematical appendix, ''IRE Trans. Inform. Theory,'' vol. 6, 59-267.</ref>
==Properties== ===Finite integral representation=== Using the fact that <math>Q_\nu (a,0) = 1</math>, the generalized Marcum Q-function can alternatively be defined as a finite integral as
: <math>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 integer values of <math>\nu = n</math>, such a representation is given by the trigonometric integral<ref name="Simon1998">M.K. Simon and M.-S. Alouini (1998). A Unified Approach to the Performance of Digital Communication over Generalized Fading Channels, ''Proceedings of the IEEE'', 86(9), 1860-1877.</ref><ref name="Annamalai2001">A. Annamalai and C. Tellambura (2001). Cauchy-Schwarz bound on the generalized Marcum-Q function with applications, ''Wireless Communications and Mobile Computing'', 1(2), 243-253.</ref>
:<math> 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.</math>
where
:<math>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>\zeta = a/b</math> is a constant.
For any real <math>\nu > 0</math>, such finite trigonometric integral is given by<ref name="Annamalai2008">A. Annamalai and C. Tellambura (2008). A Simple Exponential Integral Representation of the Generalized Marcum Q-Function Q<sub>M</sub>(''a'',''b'') for Real-Order ''M'' with Applications. ''2008 IEEE Military Communications Conference'', San Diego, CA, USA</ref>
:<math> 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.</math>
where <math>H_n(a,b)</math> is as defined before, <math>\zeta = a/b</math>, and the additional correction term is given by
:<math> 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>\nu</math>, the correction term <math>C_\nu(a,b)</math> tend to vanish.
===Monotonicity and log-concavity=== * The generalized Marcum Q-function <math>Q_\nu(a,b)</math> is strictly increasing in <math>\nu</math> and <math>a</math> for all <math>a \geq 0</math> and <math>b, \nu > 0</math>, and is strictly decreasing in <math>b</math> for all <math>a, b \geq 0</math> and <math>\nu>0.</math><ref name="Sun2010">Y. Sun, A. Baricz, and S. Zhou (2010). On the Monotonicity, Log-Concavity, and Tight Bounds of the Generalized Marcum and Nuttall Q-Functions. ''IEEE Transactions on Information Theory'', 56(3), 1166–1186, {{ISSN|0018-9448}}</ref> * The function <math>\nu \mapsto Q_\nu(a,b)</math> is log-concave on <math>[1,\infty)</math> for all <math>a , b \geq 0.</math><ref name="Sun2010"/> * The function <math>b \mapsto Q_\nu(a,b)</math> is strictly log-concave on <math>(0,\infty)</math> for all <math>a \geq 0</math> and <math>\nu > 1</math>, which implies that the generalized Marcum Q-function satisfies the new-is-better-than-used property.<ref name="Sun2008">Y. Sun and A. Baricz (2008). Inequalities for the generalized Marcum Q-function. ''Applied Mathematics and Computation'' 203(2008) 134-141.</ref> * The function <math>a \mapsto 1 - Q_\nu(a,b)</math> is log-concave on <math>[0,\infty)</math> for all <math>b, \nu > 0.</math><ref name="Sun2010"/>
===Series representation=== * The generalized Marcum Q function of order <math>\nu > 0</math> can be represented using incomplete Gamma function as<ref name="Temme1993"/><ref name="Annamalai 2009"/><ref name="Andras2011">S. Andras, A. Baricz, and Y. Sun (2011) The Generalized Marcum Q-function: An Orthogonal Polynomial Approach. ''Acta Univ. Sapientiae Mathematica'', 3(1), 60-76.</ref>
:: <math> 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>\gamma(s,x)</math> is the lower incomplete Gamma function. This is usually called the canonical representation of the <math>\nu</math>-th order generalized Marcum Q-function.
* The generalized Marcum Q function of order <math>\nu > 0</math> can also be represented using generalized Laguerre polynomials as<ref name="Andras2011"/>
::<math> 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>L_k^{(\alpha)}(\cdot)</math> is the generalized Laguerre polynomial of degree <math>k</math> and of order <math>\alpha</math>.
* The generalized Marcum Q-function of order <math>\nu > 0</math> can also be represented as Neumann series expansions<ref name="Annamalai2008"/><ref name="Annamalai 2009"/>
:: <math>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>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>\alpha</math> assumes an integer value, we have <math>I_{\alpha}(ab) = I_{-\alpha}(ab)</math>.
* For non-negative half-integer values <math>\nu = n + 1/2</math>, we have a closed form expression for the generalized Marcum Q-function as<ref name="Annamalai 2009"/><ref name="Brychkov"/>
::<math>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>\mathrm{erfc}(\cdot)</math> is the complementary error function. Since Bessel functions with half-integer parameter have finite sum expansions as<ref name="Annamalai2008"/>
::<math>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>n</math> is non-negative integer, we can exactly represent the generalized Marcum Q-function with half-integer parameter. More precisely, we have<ref name="Annamalai2008"/>
::<math>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>n</math>, where <math>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:<ref name="AbramowitzStegun1972">M. Abramowitz and I.A. Stegun (1972). [https://personal.math.ubc.ca/~cbm/aands/page_443.htm Formula 10.2.12, Modified Spherical Bessel Functions], ''Handbook of Mathematical functions'', p. 443</ref>
::<math>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>g_0(z) = z^{-1}</math>, <math>g_1(z) = -z^{-2}</math>, and <math>g_{n-1}(z) - g_{n+1}(z) = (2n+1) z^{-1} g_n(z)</math> for any integer value of <math>n</math>.
===Recurrence relation and generating function=== * Integrating by parts, we can show that generalized Marcum Q-function satisfies the following recurrence relation<ref name="Annamalai 2009">A. Annamalai, C. Tellambura and John Matyjas (2009). "A New Twist on the Generalized Marcum Q-Function ''Q''<sub>''M''</sub>(''a'', ''b'') with Fractional-Order ''M'' and its Applications". ''2009 6th IEEE Consumer Communications and Networking Conference'', 1–5, {{ISBN|978-1-4244-2308-8}}</ref><ref name="Brychkov"/>
:: <math> 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<ref name="Brychkov"/>
::<math>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>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>n</math>. The former recurrence can be used to formally define the generalized Marcum Q-function for negative <math>\nu</math>. Taking <math>Q_\infty(a,b)=1</math> and <math>Q_{-\infty}(a,b)=0</math> for <math>n = \infty</math>, we obtain the Neumann series representation of the generalized Marcum Q-function.
* The related three-term recurrence relation is given by<ref name="Temme1993"/>
::<math>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>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<ref name="Temme1993"/>
::<math>\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>Q_{\nu+1}(a,b) = Q_\nu(a,b) + \frac{1}{a} \frac{\partial}{\partial a} Q_\nu(a,b),</math> ::<math>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>Q_\nu(a,b)</math> for integral <math>\nu</math> is<ref name="Brychkov"/>
::<math>\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>|t|<1.</math>
===Symmetry relation=== * Using the two Neumann series representations, we can obtain the following symmetry relation for positive integral <math>\nu = n</math>
::<math>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>n = 1</math> we have
::<math>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<ref name="Sun2008"/> * <math> Q_\nu(0,0) = 1, </math> * <math> Q_\nu(a,0) = 1, </math> * <math> Q_\nu(a,+\infty) = 0, </math> * <math> Q_\nu(0,b) = \frac{\Gamma(\nu,b^2/2)}{\Gamma(\nu)}, </math> * <math> Q_\nu(+\infty,b) = 1, </math> * <math> Q_\infty(a,b) = 1, </math> * For <math>a=b</math>, by subtracting the two forms of Neumann series representations, we have<ref name="Brychkov">Y.A. Brychkov (2012). On some properties of the Marcum Q function. ''Integral Transforms and Special Functions'' 23(3), 177-182.</ref>
::<math>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>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>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>n</math>.
* For <math>\nu = 1/2</math>, using the basic integral definition of generalized Marcum Q-function, we have<ref name="Annamalai 2009"/><ref name="Brychkov"/>
::<math> 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>\nu=3/2</math>, we have
::<math>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>\nu = 5/2</math> we have
::<math>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>\nu</math> to be fixed and <math>ab</math> large, let <math>\zeta = a/b > 0</math>, then the generalized Marcum-Q function has the following asymptotic form<ref name="Temme1993">N.M. Temme (1993). Asymptotic and numerical aspects of the noncentral chi-square distribution. ''Computers Math. Applic.'', 25(5), 55-63.</ref>
::<math>Q_\nu(a,b) \sim \sum_{n=0}^\infty \psi_n,</math>
:where <math>\psi_n</math> is given by
::<math>\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>\phi_n</math> and <math>A_n</math> are given by
::<math>\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>A_n(\nu) = \frac{2^{-n}\Gamma(\frac{1}{2}+\nu+n)}{n!\Gamma(\frac{1}{2}+\nu-n)}.</math>
:The function <math>A_n(\nu)</math> satisfies the recursion
::<math>A_{n+1}(\nu) = - \frac{(2n+1)^2 - 4\nu^2}{8(n+1)}A_n(\nu),</math>
:for <math>n \geq 0</math> and <math>A_0(\nu)=1.</math>
* In the first term of the above asymptotic approximation, we have
::<math>\phi_0 = \frac{\sqrt{2 \pi ab}}{b-a} \mathrm{erfc}\left(\frac{b-a}{\sqrt{2}}\right).</math>
:Hence, assuming <math>b>a</math>, the first term asymptotic approximation of the generalized Marcum-Q function is<ref name="Temme1993"/>
::<math>Q_\nu(a,b) \sim \psi_0 = \left(\frac{b}{a}\right)^{\nu-\frac{1}{2}} Q(b-a),</math>
:where <math>Q(\cdot)</math> is the Gaussian Q-function. Here <math>Q_\nu(a,b) \sim 0.5</math> as <math>a \uparrow b.</math>
:For the case when <math>a > b</math>, we have<ref name="Temme1993"/>
::<math>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>Q_\nu(a,b) \sim 0.5</math> as <math>a \downarrow b.</math>
===Differentiation=== * The partial derivative of <math>Q_\nu(a,b)</math> with respect to <math>a</math> and <math>b</math> is given by<ref>W.K. Pratt (1968). Partial Differentials of Marcum's Q Function. ''Proceedings of the IEEE'', 56(7), 1220-1221.</ref><ref>R. Esposito (1968). Comment on Partial Differentials of Marcum's Q Function. ''Proceedings of the IEEE'', 56(12), 2195-2195.</ref>
::<math> \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> \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>\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>Q_\nu(a,b)</math> with respect to its arguments is given by<ref name="Brychkov"/>
::<math> \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> \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<ref name="Sun2010"/>
::<math>Q^2_\nu(a,b) > \frac{Q_{\nu-1}(a,b) + Q_{\nu+1}(a,b)}{2} > Q_{\nu-1}(a,b) Q_{\nu+1}(a,b)</math>
:for all <math>a \geq b > 0</math> and <math>\nu > 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>\nu \mapsto Q_\nu(a,b)</math> and the fact that we have closed form expression for <math>Q_\nu(a,b)</math> when <math>\nu</math> is half-integer valued.
Let <math>\lfloor x \rfloor_{0.5}</math> and <math>\lceil x \rceil_{0.5}</math> denote the pair of half-integer rounding operators that map a real <math>x</math> to its nearest left and right half-odd integer, respectively, according to the relations
:<math>\lfloor x \rfloor_{0.5} = \lfloor x - 0.5 \rfloor + 0.5</math> :<math> \lceil x \rceil_{0.5} = \lceil x + 0.5 \rceil - 0.5</math>
where <math>\lfloor x \rfloor</math> and <math>\lceil x \rceil</math> denote the integer floor and ceiling functions.
* The monotonicity of the function <math>\nu \mapsto Q_\nu(a,b)</math> for all <math>a \geq 0</math> and <math>b > 0</math> gives us the following simple bound<ref>V.M. Kapinas, S.K. Mihos, G.K. Karagiannidis (2009). On the Monotonicity of the Generalized Marcum and Nuttal Q-Functions. ''IEEE Transactions on Information Theory'', 55(8), 3701-3710.</ref><ref name="Annamalai 2009"/><ref name="Li2010">R. Li, P.Y. Kam, and H. Fu (2010). New Representations and Bounds for the Generalized Marcum Q-Function via a Geometric Approach, and an Application. ''IEEE Trans. Commun.'', 58(1), 157-169.</ref>
::<math>Q_{\lfloor\nu\rfloor_{0.5}}(a,b) < Q_\nu(a,b) < Q_{\lceil\nu\rceil_{0.5}}(a,b).</math>
:However, the relative error of this bound does not tend to zero when <math>b \to \infty</math>.<ref name="Sun2010"/> For integral values of <math>\nu = n</math>, this bound reduces to
::<math>Q_{n-0.5}(a,b) < Q_n(a,b) < Q_{n+0.5}(a,b).</math>
:A very good approximation of the generalized Marcum Q-function for integer valued <math>\nu = n</math> is obtained by taking the arithmetic mean of the upper and lower bound<ref name="Li2010"/>
::<math> 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>\nu \mapsto Q_\nu(a,b)</math> on <math>[1,\infty)</math> as<ref name="Sun2010"/>
::<math>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}},</math>
:where <math>\nu_1 = \lfloor\nu\rfloor_{0.5}</math> and <math>\nu_2 = \lceil\nu\rceil_{0.5}</math> for <math>\nu \geq 1.5</math>. The tightness of this bound improves as either <math>a</math> or <math>\nu</math> increases. The relative error of this bound converges to 0 as <math>b \to \infty</math>.<ref name="Sun2010"/> For integral values of <math>\nu = n</math>, this bound reduces to
::<math>\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)}}.</math>
===Cauchy-Schwarz bound=== Using the trigonometric integral representation for integer valued <math>\nu=n</math>, the following Cauchy-Schwarz bound can be obtained<ref name="Annamalai2001"/>
:<math>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,</math> :<math>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,</math>
where <math>\zeta = a/b >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>\zeta = a/b >0</math>, one such bound for integer valued <math>\nu = n</math> is given as<ref name="Simon2000">M.K. Simon and M.-S. Alouini (2000). Exponential-Type Bounds on the Generalized Marcum Q-Function with Application to Error Probability Analysis over Fading Channels. ''IEEE Trans. Commun.'' 48(3), 359-366.</ref><ref name="Annamalai2001"/>
:<math>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, </math> :<math>Q_n(a,b) \geq 1 - \frac{1}{2}\left[e^{-(a-b)^2/2} - e^{-(a+b)^2/2} \right], \qquad \zeta > 1.</math>
When <math>n=1</math>, the bound simplifies to give
:<math>e^{-(b+a)^2/2} \leq Q_1(a,b) \leq e^{-(b-a)^2/2}, \qquad \zeta <1, </math> :<math>1 - \frac{1}{2}\left[e^{-(a-b)^2/2} - e^{-(a+b)^2/2} \right] \leq Q_1(a,b), \qquad \zeta > 1.</math>
Another such bound obtained via Cauchy-Schwarz inequality is given as<ref name="Annamalai2001"/>
:<math>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</math> :<math>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.</math>
===Chernoff-type bound=== Chernoff-type bounds for the generalized Marcum Q-function, where <math>\nu = n</math> is an integer, is given by<ref name="Simon2000"/><ref name="Annamalai2001"/>
:<math>(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 > n(a^2+2) \\ 1 - Q_n(a,b), & b^2 < n(a^2+2) \end{array} \right.</math>
where the Chernoff parameter <math>(0 < \lambda < 1/2)</math> has optimum value <math>\lambda_0</math> of
:<math>\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 <ref name="Guo2021">H. Guo, B. Makki, M. -S. Alouini and T. Svensson, "A Semi-Linear Approximation of the First-Order Marcum Q-Function With Application to Predictor Antenna Systems," in ''IEEE Open Journal of the Communications Society'', vol. 2, pp. 273-286, 2021, doi: 10.1109/OJCOMS.2021.3056393.</ref>
:<math>\begin{align} 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{align}</math> where :<math> \begin{align} \beta_0 = \frac{a+\sqrt{a^2+2}}{2}, \end{align} </math> :<math> \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> \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<ref name="Shnidman1989">D.A. Shnidman (1989). The Calculation of the Probability of Detection and the Generalized Marcum Q-Function. ''IEEE Transactions on Information Theory,'' 35(2), 389-400.</ref>
: <math> P_N(X,Y) = Q_N(\sqrt{2NX},\sqrt{2Y}). </math>
The <math>P_N(X,Y)</math> can be interpreted as the detection probability of <math>N</math> incoherently integrated received signal samples of constant received signal-to-noise ratio, <math>X</math>, with a normalized detection threshold <math>Y</math>. In this equivalent form of Marcum Q-function, for given <math>a</math> and <math>b</math>, we have <math>X = a^2/2N</math> and <math>Y = b^2/2</math>. Many expressions exist that can represent <math>P_N(X,Y)</math>. However, the five most reliable, accurate, and efficient ones for numerical computation are given below. They are form one:<ref name="Shnidman1989"/>
: <math> 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:<ref name="Shnidman1989"/>
: <math> 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:<ref name="Shnidman1989"/>
: <math> 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:<ref name="Shnidman1989"/>
: <math> 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:<ref name="Shnidman1989"/>
: <math> 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.<ref name="Shnidman1989"/>
==Applications== The generalized Marcum Q-function can be used to represent the cumulative distribution function (cdf) of many random variables:
* If <math>X \sim \mathrm{Exp}(\lambda)</math> is an exponential distribution with rate parameter <math>\lambda</math>, then its cdf is given by <math>F_X(x) = 1 - Q_1\left(0,\sqrt{2 \lambda x}\right)</math> * If <math>X \sim \mathrm{Erlang}(k,\lambda)</math> is a Erlang distribution with shape parameter <math>k</math> and rate parameter <math>\lambda</math>, then its cdf is given by <math>F_X(x) = 1 - Q_k\left(0,\sqrt{2 \lambda x}\right)</math> * If <math>X \sim \chi^2_k</math> is a chi-squared distribution with <math>k</math> degrees of freedom, then its cdf is given by <math>F_X(x) = 1 - Q_{k/2}(0,\sqrt{x})</math> * If <math>X \sim \mathrm{Gamma}(\alpha,\beta)</math> is a gamma distribution with shape parameter <math>\alpha</math> and rate parameter <math>\beta</math>, then its cdf is given by <math>F_X(x) = 1 - Q_{\alpha}(0,\sqrt{2 \beta x})</math> * If <math>X \sim \mathrm{Weibull}(k,\lambda)</math> is a Weibull distribution with shape parameters <math>k</math> and scale parameter <math>\lambda</math>, then its cdf is given by <math>F_X(x) = 1 - Q_1 \left( 0, \sqrt{2} \left(\frac{x}{\lambda}\right)^{\frac{k}{2}} \right)</math> * If <math>X \sim \mathrm{GG}(a,d,p)</math> is a generalized gamma distribution with parameters <math>a, d, p</math>, then its cdf is given by <math>F_X(x) = 1 - Q_{\frac{d}{p}} \left( 0, \sqrt{2} \left(\frac{x}{a}\right)^{\frac{p}{2}} \right)</math> * If <math>X \sim \chi^2_k(\lambda)</math> is a non-central chi-squared distribution with non-centrality parameter <math>\lambda</math> and <math>k</math> degrees of freedom, then its cdf is given by <math>F_X(x) = 1 - Q_{k/2}(\sqrt{\lambda},\sqrt{x})</math> * If <math>X \sim \mathrm{Rayleigh}(\sigma)</math> is a Rayleigh distribution with parameter <math>\sigma</math>, then its cdf is given by <math>F_X(x) = 1 - Q_1\left(0,\frac{x}{\sigma}\right)</math> * If <math>X \sim \mathrm{Maxwell}(\sigma)</math> is a Maxwell–Boltzmann distribution with parameter <math>\sigma</math>, then its cdf is given by <math>F_X(x) = 1 - Q_{3/2}\left(0,\frac{x}{\sigma}\right)</math> * If <math>X \sim \chi_k</math> is a chi distribution with <math>k</math> degrees of freedom, then its cdf is given by <math>F_X(x) = 1 - Q_{k/2}(0,x)</math> * If <math>X \sim \mathrm{Nakagami}(m,\Omega)</math> is a Nakagami distribution with <math>m</math> as shape parameter and <math>\Omega</math> as spread parameter, then its cdf is given by <math>F_X(x) = 1 - Q_{m}\left(0,\sqrt{\frac{2m}{\Omega}}x\right)</math> * If <math>X \sim \mathrm{Rice}(\nu,\sigma)</math> is a Rice distribution with parameters <math>\nu</math> and <math>\sigma</math>, then its cdf is given by <math>F_X(x) = 1 - Q_1\left(\frac{\nu}{\sigma},\frac{x}{\sigma}\right)</math> * If <math>X \sim \chi_k(\lambda)</math> is a non-central chi distribution with non-centrality parameter <math>\lambda</math> and <math>k</math> degrees of freedom, then its cdf is given by <math>F_X(x) = 1 - Q_{k/2}(\lambda,x)</math>
==Footnotes== <references/>
== 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): ''[https://dx.doi.org/10.1109/TIT.1975.1055327 Some Integrals Involving the Q<sub>M</sub> 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. [https://mathworld.wolfram.com/MarcumQ-Function.html]
Category:Functions related to probability distributions