In statistics, the generalized Marcum Q-function of order is defined as
where and and is the modified Bessel function of first kind of order . If , the integral converges for any . 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 by, and hence named after, Jess Marcum for pulsed radars.J.I. Marcum (1960). A statistical theory of target detection by pulsed radar: mathematical appendix, IRE Trans. Inform. Theory, vol. 6, 59-267.
Properties Finite integral representation Using the fact that , the generalized Marcum Q-function can alternatively be defined as a finite integral as
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 , such a representation is given by the trigonometric integralM.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.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.
where
and the ratio is a constant.
For any real , such finite trigonometric integral is given byA. Annamalai and C. Tellambura (2008). A Simple Exponential Integral Representation of the Generalized Marcum Q-Function QM(a,b) for Real-Order M with Applications. 2008 IEEE Military Communications Conference, San Diego, CA, USA
where is as defined before, , and the additional correction term is given by
For integer values of , the correction term tend to vanish.
Monotonicity and log-concavity The generalized Marcum Q-function is strictly increasing in and for all and , and is strictly decreasing in for all and 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, The function is log-concave on for all The function is strictly log-concave on for all and , which implies that the generalized Marcum Q-function satisfies the new-is-better-than-used property.Y. Sun and A. Baricz (2008). Inequalities for the generalized Marcum Q-function. Applied Mathematics and Computation 203(2008) 134-141. The function is log-concave on for all
Series representation The generalized Marcum Q function of order can be represented using incomplete Gamma function asS. Andras, A. Baricz, and Y. Sun (2011) The Generalized Marcum Q-function: An Orthogonal Polynomial Approach. Acta Univ. Sapientiae Mathematica, 3(1), 60-76.
where is the lower incomplete Gamma function. This is usually called the canonical representation of the -th order generalized Marcum Q-function.
The generalized Marcum Q function of order can also be represented using generalized Laguerre polynomials as
where is the generalized Laguerre polynomial of degree and of order .
The generalized Marcum Q-function of order can also be represented as Neumann series expansions
where the summations are in increments of one. Note that when assumes an integer value, we have .
For non-negative half-integer values , we have a closed form expression for the generalized Marcum Q-function as
where is the complementary error function. Since Bessel functions with half-integer parameter have finite sum expansions as
where is non-negative integer, we can exactly represent the generalized Marcum Q-function with half-integer parameter. More precisely, we have
for non-negative integers , where 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:M. Abramowitz and I.A. Stegun (1972). Formula 10.2.12, Modified Spherical Bessel Functions, Handbook of Mathematical functions, p. 443
where , , and for any integer value of .
Recurrence relation and generating function Integrating by parts, we can show that generalized Marcum Q-function satisfies the following recurrence relationA. Annamalai, C. Tellambura and John Matyjas (2009). "A New Twist on the Generalized Marcum Q-Function QM(a, b) with Fractional-Order M and its Applications". 2009 6th IEEE Consumer Communications and Networking Conference, 1–5,
The above formula is easily generalized as
for positive integer . The former recurrence can be used to formally define the generalized Marcum Q-function for negative . Taking and for , we obtain the Neumann series representation of the generalized Marcum Q-function.
The related three-term recurrence relation is given by
where
We can eliminate the occurrence of the Bessel function to give the third order recurrence relation
Another recurrence relationship, relating it with its derivatives, is given by
The ordinary generating function of for integral is
where
Symmetry relation Using the two Neumann series representations, we can obtain the following symmetry relation for positive integral
In particular, for we have
Special values Some specific values of Marcum-Q function are
For , by subtracting the two forms of Neumann series representations, we haveY.A. Brychkov (2012). On some properties of the Marcum Q function. Integral Transforms and Special Functions 23(3), 177-182.
which when combined with the recursive formula gives
for any non-negative integer .
For , using the basic integral definition of generalized Marcum Q-function, we have
For , we have
For we have
Asymptotic forms Assuming to be fixed and large, let , then the generalized Marcum-Q function has the following asymptotic formN.M. Temme (1993). Asymptotic and numerical aspects of the noncentral chi-square distribution. Computers Math. Applic., 25(5), 55-63.
where is given by
The functions and are given by
The function satisfies the recursion
for and
In the first term of the above asymptotic approximation, we have
Hence, assuming , the first term asymptotic approximation of the generalized Marcum-Q function is
where is the Gaussian Q-function. Here as
For the case when , we have
Here too as
Differentiation The partial derivative of with respect to and is given byW.K. Pratt (1968). Partial Differentials of Marcum's Q Function. Proceedings of the IEEE, 56(7), 1220-1221.R. Esposito (1968). Comment on Partial Differentials of Marcum's Q Function. Proceedings of the IEEE, 56(12), 2195-2195.
We can relate the two partial derivatives as
The n-th partial derivative of with respect to its arguments is given by
Inequalities The generalized Marcum-Q function satisfies a Turán-type inequality
for all and .
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 and the fact that we have closed form expression for when is half-integer valued.
Let and denote the pair of half-integer rounding operators that map a real to its nearest left and right half-odd integer, respectively, according to the relations
where and denote the integer floor and ceiling functions.
The monotonicity of the function for all and gives us the following simple boundV.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.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.
However, the relative error of this bound does not tend to zero when . For integral values of , this bound reduces to
A very good approximation of the generalized Marcum Q-function for integer valued is obtained by taking the arithmetic mean of the upper and lower bound
A tighter bound can be obtained by exploiting the log-concavity of on as
where and for . The tightness of this bound improves as either or increases. The relative error of this bound converges to 0 as . For integral values of , this bound reduces to
Cauchy-Schwarz bound Using the trigonometric integral representation for integer valued , the following Cauchy-Schwarz bound can be obtained
where .
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 , one such bound for integer valued is given asM.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.
When , the bound simplifies to give
Another such bound obtained via Cauchy-Schwarz inequality is given as
Chernoff-type bound Chernoff-type bounds for the generalized Marcum Q-function, where is an integer, is given by
where the Chernoff parameter has optimum value of
Semi-linear approximation The first-order Marcum-Q function can be semi-linearly approximated by 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.
where
and
Equivalent forms for efficient computation It is convenient to re-express the Marcum Q-function asD.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.
The can be interpreted as the detection probability of incoherently integrated received signal samples of constant received signal-to-noise ratio, , with a normalized detection threshold . In this equivalent form of Marcum Q-function, for given and , we have and . Many expressions exist that can represent . However, the five most reliable, accurate, and efficient ones for numerical computation are given below. They are form one:
form two:
form three:
form four:
and form five:
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 is an exponential distribution with rate parameter , then its cdf is given by If is a Erlang distribution with shape parameter and rate parameter , then its cdf is given by If is a chi-squared distribution with degrees of freedom, then its cdf is given by If is a gamma distribution with shape parameter and rate parameter , then its cdf is given by If is a Weibull distribution with shape parameters and scale parameter , then its cdf is given by If is a generalized gamma distribution with parameters , then its cdf is given by If is a non-central chi-squared distribution with non-centrality parameter and degrees of freedom, then its cdf is given by If is a Rayleigh distribution with parameter , then its cdf is given by If is a Maxwell–Boltzmann distribution with parameter , then its cdf is given by If is a chi distribution with degrees of freedom, then its cdf is given by If is a Nakagami distribution with as shape parameter and as spread parameter, then its cdf is given by If is a Rice distribution with parameters and , then its cdf is given by If is a non-central chi distribution with non-centrality parameter and degrees of freedom, then its cdf is given by
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