Probability density function


Cumulative distribution function


Parameters  shape (real) spread (real) 

Support  
CDF  
Mean  
Median  No simple closed form 
Mode  
Variance 
The Nakagami distribution or the Nakagamim distribution is a probability distribution related to the gamma distribution. It has two parameters: a shape parameter and a second parameter controlling spread, .
Contents
Characterization[edit]
Its probability density function (pdf) is^{[1]}
Its cumulative distribution function is^{[1]}
where P is the incomplete gamma function (regularized).
Parameter estimation[edit]
The parameters and are^{[2]}
and
An alternative way of fitting the distribution is to reparametrize 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 NewtonRaphson method:
and
It is reported by authors^{[who?]} that modelling data with Nakagami distribution and estimating parameters by above mention method results in better performance for low data regime compared to moments based methods.
Generation[edit]
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 :
 .
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 Chidistributed random variable as below:
 Currently, the more efficient generation method is provided in.^{[4]}
History and applications[edit]
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]}
Related distributions[edit]
 Restricting m to the unit interval (q = m; 0 < q < 1) defines the Nakagamiq distribution, also known as Hoyt distribution.^{[7]}^{[8]}^{[9]}
"The radius around the true mean in a bivariate normal random variable, rewritten in polar coordinates (radius and angle), follows a Hoyt distribution. Equivalently, the modulus of a complex normal random variable does."^{[10]}
References[edit]
This article needs additional citations for verification. (April 2013) (Learn how and when to remove this template message) 
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (April 2013) 
 ^ ^{a} ^{b} Laurenson, Dave (1994). "Nakagami Distribution". Indoor Radio Channel Propagation Modelling by Ray Tracing Techniques. Retrieved 20070804.
 ^ R. Kolar, R. Jirik, J. Jan (2004) "Estimator Comparison of the Nakagamim 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 Nakagamim Distribution". International Conference on Communications, Devices and Intelligent Systems (CODIS), 2012: 9–12.
 ^ Luengo, D.; Martino, L. "Almost rejectionless sampling from Nakagamim distributions (m≥1)". Electronics Letters 48 (24): 1559–1561. doi:10.1049/el.2012.3513.
 ^ Nakagami, M. (1960) "The mDistribution, 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 336. Pergamon Press.
 ^ Parsons, J. D. (1992) The Mobile Radio Propagation Channel. New York: Wiley.
 ^ "Nakagamiq (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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