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Nakagami
Probability density function
Cumulative distribution function
Parameters	${\displaystyle m\text{ or }\mu \ge 0.5}$shape (real) ${\displaystyle \mathrm{\Omega }\text{ or }\omega >0}$scale (real)
Support	${\displaystyle x>0}$
PDF	${\displaystyle \frac{2m^m}{\mathrm{\Gamma }(m){\mathrm{\Omega }}^m}{x}^{2m-1}\exp \left(-\frac{m}{\mathrm{\Omega }}{x}^2\right)}$
CDF	${\displaystyle \frac{\gamma \left(m,\frac{m}{\mathrm{\Omega }}{x}^2\right)}{\mathrm{\Gamma }(m)}}$
Mean	${\displaystyle \frac{\mathrm{\Gamma }(m+\frac{1}{2})}{\mathrm{\Gamma }(m)}{\left(\frac{\mathrm{\Omega }}{m}\right)}^{1/2}}$
Median	No simple closed form
Mode	${\displaystyle {\left(\frac{(2m-1)\mathrm{\Omega }}{2m}\right)}^{1/2}}$
Variance	${\displaystyle \mathrm{\Omega }\left(1-\frac{1}{m}{\left(\frac{\mathrm{\Gamma }(m+\frac{1}{2})}{\mathrm{\Gamma }(m)}\right)}^2\right)}$

TheNakagami distribution or theNakagami-m distribution is aprobability distribution related to thegamma distribution. The family of Nakagami distributions has two parameters: ashape parameter${\displaystyle m\ge 1/2}$ and ascale parameter${\displaystyle \mathrm{\Omega }>0}$.It is used to model physical phenomena such as those found in medical ultrasound imaging, communications engineering, meteorology, hydrology, multimedia, and seismology.

Itsprobability density function (pdf) is^[1]

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

where${\displaystyle m\ge 1/2}$ and${\displaystyle \mathrm{\Omega }>0}$.

Itscumulative distribution function (CDF) is^[1]

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

whereP is the regularized (lower)incomplete gamma function.

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

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

and

${\displaystyle \mathrm{\Omega }=\mathrm{E}[X^2].}$

No closed form solution exists for themedian of this distribution, although special cases do exist, such as${\displaystyle \sqrt{\mathrm{\Omega }\ln (2)}}$ whenm = 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${\displaystyle \mathrm{\Omega }}$ asσ = Ω/m.^[3]

Givenindependent observations$X_1=x_1,\dots ,X_n=x_n$ from the Nakagami distribution, thelikelihood function is

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

Its logarithm is

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

Therefore

${\displaystyle \begin{array}{r}\frac{\mathrm{\partial }\ell }{\mathrm{\partial }\sigma }=\frac{-nm\sigma +\sum _{i=1}^n{x}_i^2}{{\sigma }^2}\text{and}\frac{\mathrm{\partial }\ell }{\mathrm{\partial }m}=-n\frac{{\mathrm{\Gamma }}^{\prime }(m)}{\mathrm{\Gamma }(m)}-n\log \sigma +2\sum _{i=1}^n\log x_i.\end{array}}$

These derivatives vanish only when

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

and the value ofm for which the derivative with respect tom vanishes is found by numerical methods including theNewton–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 theequivariance of maximum-likelihood estimation, a maximum likelihood estimate for Ω is obtained as well.

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

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

Alternatively, the Nakagami distribution${\displaystyle f(y;m,\mathrm{\Omega })}$ can be generated from thechi 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.

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

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.

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.^[4] It has been used to model attenuation ofwireless signalstraversing multiple paths^[5] and to study the impact offading channels on wireless communications.^[6]

• Restrictingm to the unit interval (q =m; 0 <q < 1)^{[dubious –discuss]} defines theNakagami-q distribution, also known asHoyt distribution, first studied by R.S. Hoyt in the 1940s.^[7][8][9] In particular, theradius around the true mean in abivariate normal random variable, re-written inpolar coordinates (radius and angle), follows a Hoyt distribution. Equivalently, themodulus of acomplex normal random variable also does.
• With 2m =k, the Nakagami distribution gives a scaledchi distribution.
• With${\displaystyle m=\frac{1}{2}}$, the Nakagami distribution gives a scaledhalf-normal distribution.
• A Nakagami distribution is a particular form ofgeneralized gamma distribution, withp = 2 andd = 2m.

• Gamma distribution
• Modified half-normal distribution
• Sub-Gaussian distribution

• Continuous distributions

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