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 , and hence named after, by 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 a 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]