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.
Properties
Finite integral representation
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 integral value of , such a representation is given by the trigonometric integral
where
and the ratio is a constant.
For any real , such finite trigonometric integral is given by
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
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.
The function is log-concave on for all
Series representation
The generalized Marcum Q function of order can be represented using incomplete Gamma function as
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:
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 relation
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 have
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 form