Marcum Q-function

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.

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 the limits of the integral are independent of the arguments of the function, and that the limits are finite, 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
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
The generalized Marcum Q function of order can also be represented using generalized Laguerre polynomials as
The generalized Marcum Q-function of order can also be represented as Neumann series expansions
For non-negative half-integer values, we have a closed form expression for the generalized Marcum Q-function as

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
The related three-term recurrence relation is given by
Another recurrence relationship, relating it with its derivatives, is given by
The ordinary generating function of for integral is

Symmetry relation

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

Special values

Some specific values of Marcum-Q function are

For, by subtracting the two forms of Neumann series representations, we have
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
In the first term of the above asymptotic approximation, we have

Differentiation

The partial derivative of with respect to and is given by
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

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 bound
A tighter bound can be obtained by exploiting the log-concavity of on as

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 as
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
where
and

Equivalent forms for efficient computation

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