Nakagami distribution

Its probability density function (pdf) is[1]

${\displaystyle f(x;\,m,\Omega )={\frac {2m^{m}}{\Gamma (m)\Omega ^{m}}}x^{2m-1}\exp \left(-{\frac {m}{\Omega }}x^{2}\right),\forall x\geq 0.}$

where ${\displaystyle (m\geq 1/2,{\text{ and }}\Omega >0)}$

Its cumulative distribution function is[1]

${\displaystyle F(x;\,m,\Omega )=P\left(m,{\frac {m}{\Omega }}x^{2}\right)}$

where P is the incomplete gamma function (regularized).

The parameters ${\displaystyle m}$ and ${\displaystyle \Omega }$ are[2]

${\displaystyle m={\frac {\left(\operatorname {E} \left[X^{2}\right]\right)^{2}}{\operatorname {Var} \left[X^{2}\right]}},}$

and

${\displaystyle \Omega =\operatorname {E} \left[X^{2}\right].}$

An alternative way of fitting the distribution is to re-parametrize ${\displaystyle \Omega }$ and m as σ = Ω/m and m.[3]

Given independent observations ${\textstyle X_{1}=x_{1},\ldots ,X_{n}=x_{n}}$ from the Nakagami distribution, the likelihood function is

${\displaystyle L(\sigma ,m)=\left({\frac {2}{\Gamma (m)\sigma ^{m}}}\right)^{n}\left(\prod _{i=1}^{n}x_{i}\right)^{2m-1}\exp \left(-{\frac {\sum _{i=1}^{n}x_{i}^{2}}{\sigma }}\right).}$

Its logarithm is

${\displaystyle \ell (\sigma ,m)=\log L(\sigma ,m)=-n\log \Gamma (m)-nm\log \sigma +(2m-1)\sum _{i=1}^{n}\log x_{i}-{\frac {\sum _{i=1}^{n}x_{i}^{2}}{\sigma }}.}$

Therefore

${\displaystyle {\begin{aligned}{\frac {\partial \ell }{\partial \sigma }}={\frac {-nm\sigma +\sum _{i=1}^{n}x_{i}^{2}}{\sigma ^{2}}}\quad {\text{and}}\quad {\frac {\partial \ell }{\partial m}}=-n{\frac {\Gamma '(m)}{\Gamma (m)}}-n\log \sigma +2\sum _{i=1}^{n}\log x_{i}.\end{aligned}}}$

These derivatives vanish only when

${\displaystyle \sigma ={\frac {\sum _{i=1}^{n}x_{i}^{2}}{nm}}}$

and the value of m for which the derivative with respect to m vanishes is found by numerical methods including the Newton–Raphson 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,σ). Because of the equivariance of maximum-likelihood estimation, one then obtains the MLE for Ω as well.

The Nakagami distribution is related to the gamma distribution. In particular, given a random variable ${\displaystyle Y\,\sim {\textrm {Gamma}}(k,\theta )}$, it is possible to obtain a random variable ${\displaystyle X\,\sim {\textrm {Nakagami}}(m,\Omega )}$, by setting ${\displaystyle k=m}$, ${\displaystyle \theta =\Omega /m}$, and taking the square root of ${\displaystyle Y}$:

${\displaystyle X={\sqrt {Y}}.\,}$

Alternatively, the Nakagami distribution ${\displaystyle f(y;\,m,\Omega )}$ can be generated from the chi distribution with parameter ${\displaystyle k}$ set to ${\displaystyle 2m}$ and then following it by a scaling transformation of random variables. That is, a Nakagami random variable ${\displaystyle X}$ is generated by a simple scaling transformation on a Chi-distributed random variable ${\displaystyle Y\sim \chi (2m)}$ as below.

${\displaystyle X={\sqrt {(\Omega /2m)Y}}.}$

For a Chi-distribution, the degrees of freedom ${\displaystyle 2m}$ must be an integer, but for Nakagami the ${\displaystyle m}$ 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.

Finally, there is also a more efficient generation method using efficient rejection-sampling.[4]

The Nakagami distribution is relatively new, being first proposed in 1960.[5] It has been used to model attenuation of wireless signals traversing multiple paths [6] and to study the impact of fading channels on wireless communications.[7]

• Restricting m to the unit interval (q = m; 0 < q < 1) defines the Nakagami-q distribution, also known as Hoyt distribution.[8][9][10]

"The radius around the true mean in a bivariate normal random variable, re-written in polar coordinates (radius and angle), follows a Hoyt distribution. Equivalently, the modulus of a complex normal random variable does."