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.[1]

Properties

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Finite integral representation

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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 integral[2][3]

 

where

 

and the ratio   is a constant.

For any real  , such finite trigonometric integral is given by[4]

 

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

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  • The generalized Marcum Q-function   is strictly increasing in   and   for all   and  , and is strictly decreasing in   for all   and  [5]
  • The function   is log-concave on   for all  [5]
  • 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.[6]
  • The function   is log-concave on   for all  [5]

Series representation

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  • The generalized Marcum Q function of order   can be represented using incomplete Gamma function as[7][8][9]
 
where   is the lower incomplete Gamma function. This is usually called the canonical representation of the  -th order generalized Marcum Q-function.
 
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[4][8]
 
 
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[8][10]
 
where   is the complementary error function. Since Bessel functions with half-integer parameter have finite sum expansions as[4]
 
where   is non-negative integer, we can exactly represent the generalized Marcum Q-function with half-integer parameter. More precisely, we have[4]
 
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:[11]
 
where  ,  , and   for any integer value of  .

Recurrence relation and generating function

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  • Integrating by parts, we can show that generalized Marcum Q-function satisfies the following recurrence relation[8][10]
 
  • The above formula is easily generalized as[10]
 
 
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[7]
 
where
 
We can eliminate the occurrence of the Bessel function to give the third order recurrence relation[7]
 
  • Another recurrence relationship, relating it with its derivatives, is given by
 
 
  • The ordinary generating function of   for integral   is[10]
 
where  

Symmetry relation

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  • Using the two Neumann series representations, we can obtain the following symmetry relation for positive integral  
 
In particular, for   we have
 

Special values

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Some specific values of Marcum-Q function are[6]

  •  
  •  
  •  
  •  
  •  
  •  
  • For  , by subtracting the two forms of Neumann series representations, we have[10]
 
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[8][10]
 
  • For  , we have
 
  • For   we have
 

Asymptotic forms

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  • Assuming   to be fixed and   large, let  , then the generalized Marcum-Q function has the following asymptotic form[7]
 
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[7]
 
where   is the Gaussian Q-function. Here   as  
For the case when  , we have[7]
 
Here too   as  

Differentiation

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  • The partial derivative of   with respect to   and   is given by[12][13]
 
 
We can relate the two partial derivatives as
 
  • The n-th partial derivative of   with respect to its arguments is given by[10]
 
 

Inequalities

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for all   and  .

Bounds

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Based on monotonicity and log-concavity

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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 bound[14][8][15]
 
However, the relative error of this bound does not tend to zero when  .[5] 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[15]
 
  • A tighter bound can be obtained by exploiting the log-concavity of   on   as[5]
 
where   and   for  . The tightness of this bound improves as either   or   increases. The relative error of this bound converges to 0 as  .[5] For integral values of  , this bound reduces to
 

Cauchy-Schwarz bound

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Using the trigonometric integral representation for integer valued  , the following Cauchy-Schwarz bound can be obtained[3]

 
 

where  .

Exponential-type bounds

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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 as[16][3]

 
 

When  , the bound simplifies to give

 
 

Another such bound obtained via Cauchy-Schwarz inequality is given as[3]

 
 

Chernoff-type bound

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Chernoff-type bounds for the generalized Marcum Q-function, where   is an integer, is given by[16][3]

 

where the Chernoff parameter   has optimum value   of

 

Semi-linear approximation

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The first-order Marcum-Q function can be semi-linearly approximated by [17]

 

where

 
 

and

 

Equivalent forms for efficient computation

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It is convenient to re-express the Marcum Q-function as[18]

 

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:[18]

 

form two:[18]

 

form three:[18]

 

form four:[18]

 

and form five:[18]

 

Among these five form, the second form is the most robust.[18]

Applications

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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  

Footnotes

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  1. ^ J.I. Marcum (1960). A statistical theory of target detection by pulsed radar: mathematical appendix, IRE Trans. Inform. Theory, vol. 6, 59-267.
  2. ^ 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.
  3. ^ a b c d e 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.
  4. ^ a b c d A. 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
  5. ^ a b c d e f g 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
  6. ^ a b Y. Sun and A. Baricz (2008). Inequalities for the generalized Marcum Q-function. Applied Mathematics and Computation 203(2008) 134-141.
  7. ^ a b c d e f N.M. Temme (1993). Asymptotic and numerical aspects of the noncentral chi-square distribution. Computers Math. Applic., 25(5), 55-63.
  8. ^ a b c d e f A. Annamalai, C. Tellambura and John Matyjas (2009). "A New Twist on the Generalized Marcum Q-Function QM(ab) with Fractional-Order M and its Applications". 2009 6th IEEE Consumer Communications and Networking Conference, 1–5, ISBN 978-1-4244-2308-8
  9. ^ a b 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.
  10. ^ a b c d e f g Y.A. Brychkov (2012). On some properties of the Marcum Q function. Integral Transforms and Special Functions 23(3), 177-182.
  11. ^ M. Abramowitz and I.A. Stegun (1972). Formula 10.2.12, Modified Spherical Bessel Functions, Handbook of Mathematical functions, p. 443
  12. ^ W.K. Pratt (1968). Partial Differentials of Marcum's Q Function. Proceedings of the IEEE, 56(7), 1220-1221.
  13. ^ R. Esposito (1968). Comment on Partial Differentials of Marcum's Q Function. Proceedings of the IEEE, 56(12), 2195-2195.
  14. ^ 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.
  15. ^ a b 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.
  16. ^ a b 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.
  17. ^ 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.
  18. ^ a b c d e f g 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.

References

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  • 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 QM 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]