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 m ≥ 1 / 2 {\displaystyle m\geq 1/2} and a scale parameter Ω > 0 {\displaystyle \Omega >0} . It is used to model physical phenomena such as those found in medical ultrasound imaging, communications engineering, meteorology, hydrology, multimedia, and seismology.
Characterization Its probability density function (pdf) is
f ( x ; m , Ω ) = 2 m m Γ ( m ) Ω m x 2 m − 1 exp ( − m Ω x 2 ) for x ≥ 0. {\displaystyle f(x;\,m,\Omega )={\frac {2m^{m}}{\Gamma (m)\Omega ^{m}}}x^{2m-1}\exp \left(-{\frac {m}{\Omega }}x^{2}\right){\text{ for }}x\geq 0.}
where m ≥ 1 / 2 {\displaystyle m\geq 1/2} and Ω > 0 {\displaystyle \Omega >0} . Its cumulative distribution function (CDF) is
F ( x ; m , Ω ) = γ ( m , m Ω x 2 ) Γ ( m ) = P ( m , m Ω x 2 ) {\displaystyle F(x;\,m,\Omega )={\frac {\gamma \left(m,{\frac {m}{\Omega }}x^{2}\right)}{\Gamma (m)}}=P\left(m,{\frac {m}{\Omega }}x^{2}\right)}
where P is the regularized (lower) incomplete gamma function.
Parameterization The parameters m {\displaystyle m} and Ω {\displaystyle \Omega } are
m = ( E [ X 2 ] ) 2 Var [ X 2 ] , {\displaystyle m={\frac {\left(\operatorname {E} [X^{2}]\right)^{2}}{\operatorname {Var} [X^{2}]}},}
and
Ω = E [ X 2 ] . {\displaystyle \Omega =\operatorname {E} [X^{2}].}
No closed form solution exists for the median of this distribution, although special cases do exist, such as Ω ln ( 2 ) {\displaystyle {\sqrt {\Omega \ln(2)}}} when m = 1. For practical purposes the median would have to be calculated as the 50th-percentile of the observations.
Parameter estimation An alternative way of fitting the distribution is to re-parametrize Ω {\displaystyle \Omega } as σ = Ω/m. Given independent observations X 1 = x 1 , … , X n = x n {\textstyle X_{1}=x_{1},\ldots ,X_{n}=x_{n}} from the Nakagami distribution, the likelihood function is
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