Nakagami distribution

Nakagami
	Probability density function
	Cumulative distribution function
Parameters	shape (real); spread (real)
Support
PDF
CDF
Mean
Median	No simple closed form
Mode
Variance

The Nakagami distribution or the Nakagami-m distribution is a probability distribution related to the gamma distribution. It has two parameters: a shape parameter $m\geq 1/2$ and a second parameter controlling spread $\Omega >0$ .

Characterization

Its probability density function (pdf) is^[1]

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.~~(m\geq 1/2,{\text{ and }}\Omega >0)

Its cumulative distribution function is^[1]

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

where P is the incomplete gamma function (regularized).

Parameter estimation

The parameters $m$ and $\Omega$ are^[2]

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

and

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

An alternative way of fitting the distribution is to re-parametrize $\Omega$ and m as σ = Ω/m and m.^[3] Then, by taking the derivative of log likelihood with respect to each of the new parameters, the following equations are obtained and these can be solved using the Newton-Raphson method:

{\displaystyle \Gamma(m)= \frac{x^{2m}}{\sigma^m}, }

and

\sigma= \frac{x^2}{m}

Generation

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

X = \sqrt{Y} \,

.

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

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

For a Chi-distribution, the degrees of freedom $2m$ must be an integer, but for Nakagami the $m$ 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]

History and applications

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].

Related distributions

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."^[11]

References

1 2 Laurenson, Dave (1994). "Nakagami Distribution". Indoor Radio Channel Propagation Modelling by Ray Tracing Techniques. Retrieved 2007-08-04.
↑ R. Kolar, R. Jirik, J. Jan (2004) "Estimator Comparison of the Nakagami-m Parameter and Its Application in Echocardiography", Radioengineering, 13 (1), 8–12
↑ Mitra, Rangeet; Mishra, Amit Kumar; Choubisa, Tarun (2012). "Maximum Likelihood Estimate of Parameters of Nakagami-m Distribution". International Conference on Communications, Devices and Intelligent Systems (CODIS), 2012: 9–12.
↑ Luengo, D.; Martino, L. "Almost rejectionless sampling from Nakagami-m distributions (m≥1)". Electronics Letters. 48 (24): 1559–1561. doi:10.1049/el.2012.3513.
↑ Nakagami, M. (1960) "The m-Distribution, a general formula of intensity of rapid fading". In William C. Hoffman, editor, Statistical Methods in Radio Wave Propagation: Proceedings of a Symposium held June 18–20, 1958, pp 3-36. Pergamon Press.
↑ Parsons, J. D. (1992) The Mobile Radio Propagation Channel. New York: Wiley.
↑ Ramon Sanchez-Iborra; Maria-Dolores Cano; Joan Garcia-Haro (2013). "Performance evaluation of QoE in VoIP traffic under fading channels". World Congress on Computer and Information Technology (WCCIT).
↑ "Nakagami-q (Hoyt) distribution function with applications". doi:10.1049/el:20093427.
↑ "HoytDistribution".
↑ "NakagamiDistribution".
↑ Daniel Wollschlaeger. "The Hoyt Distribution (Documentation for R package 'shotGroups' version 0.6.2)".

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[dl-1] 1 2 Laurenson, Dave (1994). "Nakagami Distribution". Indoor Radio Channel Propagation Modelling by Ray Tracing Techniques. Retrieved 2007-08-04.

[2] R. Kolar, R. Jirik, J. Jan (2004) "Estimator Comparison of the Nakagami-m Parameter and Its Application in Echocardiography", Radioengineering, 13 (1), 8–12

[paraest-3] Mitra, Rangeet; Mishra, Amit Kumar; Choubisa, Tarun (2012). "Maximum Likelihood Estimate of Parameters of Nakagami-m Distribution". International Conference on Communications, Devices and Intelligent Systems (CODIS), 2012: 9–12.

[4] Luengo, D.; Martino, L. "Almost rejectionless sampling from Nakagami-m distributions (m≥1)". Electronics Letters. 48 (24): 1559–1561. doi:10.1049/el.2012.3513.

[5] Nakagami, M. (1960) "The m-Distribution, a general formula of intensity of rapid fading". In William C. Hoffman, editor, Statistical Methods in Radio Wave Propagation: Proceedings of a Symposium held June 18–20, 1958, pp 3-36. Pergamon Press.

[6] Parsons, J. D. (1992) The Mobile Radio Propagation Channel. New York: Wiley.

[7] Ramon Sanchez-Iborra; Maria-Dolores Cano; Joan Garcia-Haro (2013). "Performance evaluation of QoE in VoIP traffic under fading channels". World Congress on Computer and Information Technology (WCCIT).

[8] "Nakagami-q (Hoyt) distribution function with applications". doi:10.1049/el:20093427.

[9] "HoytDistribution".

[10] "NakagamiDistribution".

[11] Daniel Wollschlaeger. "The Hoyt Distribution (Documentation for R package 'shotGroups' version 0.6.2)".

Probability distributions
List
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped