The Nakagami distribution or the Nakagami-m distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter and a scale parameter . It is used to model physical phenomena such as those found in medical ultrasound imaging, communications engineering, meteorology, hydrology, multimedia, and seismology.
Its probability density function (pdf) is
where and .
Its cumulative distribution function (CDF) is
where P is the regularized (lower) incomplete gamma function.
The parameters and are
and
No closed form solution exists for the median of this distribution, although special cases do exist, such as when m = 1. For practical purposes the median would have to be calculated as the 50th-percentile of the observations.
An alternative way of fitting the distribution is to re-parametrize as à= é/m.
Given independent observations from the Nakagami distribution, the likelihood function is
Its logarithm is
Therefore
These derivatives vanish only when
and the value of m for which the derivative with respect to m vanishes is found by numerical methods including the NewtonâÂÂRaphson method.
It can be shown that at the critical point a global maximum is attained, so the critical point is the maximum-likelihood estimate of (m,ÃÂ). Because of the equivariance of maximum-likelihood estimation, a maximum likelihood estimate for é is obtained as well.
The Nakagami distribution is related to the gamma distribution. In particular, given a random variable , it is possible to obtain a random variable , by setting , , and taking the square root of :
Alternatively, the Nakagami distribution can be generated from the chi distribution with parameter set to and then following it by a scaling transformation of random variables. That is, a Nakagami random variable is generated by a simple scaling transformation on a chi-distributed random variable as below.
For a chi-distribution, the degrees of freedom must be an integer, but for Nakagami the can be any real number greater than 1/2. This is the critical difference and accordingly, Nakagami-m is viewed as a generalization of chi-distribution, similar to a gamma distribution being considered as a generalization of chi-squared distributions.
The Nakagami distribution is relatively new, being first proposed in 1960 by Minoru Nakagami as a mathematical model for small-scale fading in long-distance high-frequency radio wave propagation. It has been used to model attenuation of wireless signals traversing multiple paths and to study the impact of fading channels on wireless communications.