In statistics, the generalized Marcum Q-function of order
is the modified Bessel function of first kind of order
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.
, and hence named after, by Jess Marcum for pulsed radars.
, 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
, such finite trigonometric integral is given by[4] where
, and the additional correction term is given by For integer values of
tend to vanish.
Some specific values of Marcum-Q function are[6] 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
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
denote the integer floor and ceiling functions.
Using the trigonometric integral representation for integer valued
, the following Cauchy-Schwarz bound can be obtained[3] where
For analytical purpose, it is often useful to have bounds in simple exponential form, even though they may not be the tightest bounds achievable.
, one such bound for integer valued
, the bound simplifies to give Another such bound obtained via Cauchy-Schwarz inequality is given as[3] Chernoff-type bounds for the generalized Marcum Q-function, where
is an integer, is given by[16][3] where the Chernoff parameter
of The first-order Marcum-Q function can be semi-linearly approximated by [17] where and 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
Many expressions exist that can represent
However, the five most reliable, accurate, and efficient ones for numerical computation are given below.
[18] The generalized Marcum Q-function can be used to represent the cumulative distribution function (cdf) of many random variables: