Nakagami Distribution with Heavy Tails and Applications to Mining Engineering Data
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Nakagami Distribution with Heavy Tails and Applications to Mining Engineering Data Jimmy Reyes1 · Mario A. Rojas1 · Osvaldo Venegas2 · Héctor W. Gómez1
© Grace Scientific Publishing 2020
Abstract In this paper we introduce a new extension of the Nakagami distribution. This new distribution is obtained by the quotient of two independent random variables. The quotient consists of a Nakagami distribution divided by a power of the uniform distribution in (0,1). Thus the new distribution has a heavier tail than the Nakagami distribution. In this study we obtain the density function and some important properties for making the inference, such as estimators of moment and maximum likelihood. We examine two sets of real data from the mining industry which show the usefulness of the new model in analyses with high kurtosis. Keywords Kurtosis · Maximum likelihood estimator · Slash-Nakagami distribution
1 Introduction A distribution which is related to the normal distribution is the slash distribution. It is represented as the quotient between two independent random variables one normal (N) and the other a power of the uniform distribution (U). Thus we say that S has a slash distribution if:
* Osvaldo Venegas [email protected] Jimmy Reyes [email protected] Mario A. Rojas [email protected] Héctor W. Gómez [email protected] 1
Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta, Chile
2
Departamento de Ciencias Matemáticas y Físicas, Facultad de Ingneniería, Universidad Católica de Temuco, Temuco, Chile
13
Vol.:(0123456789)
55
Page 2 of 20
Journal of Statistical Theory and Practice
Z
S=
U
1 q
,
q > 0,
(2020) 14:55
(1)
where Z ∼ N(0, 1) , U ∼ U(0, 1) , Z is independent of U (see Johnson et al.[7]) and q is the kurtosis parameter. This distribution has heavier tails than normal distribution, i.e. it has greater kurtosis. Properties of this family are discussed by Rogers and Tukey[16] and Mosteller and Tukey[11]. Location and scale maximum likelihood estimators are discussed in Kafadar[8]. Wang and Genton[17] presented a multivariate version of the slash distribution and a skew multivariate version. Gómez et al.[4] and Gómez and Venegas[5] extended the slash distribution using the family of univariate and multivariate elliptical distributions. Various works have used the slash methodology to extend some distributions with positive support, such as Gómez et al.[3] who extended the Birnbaum–Saunders distribution, Olmos et al.[13, 14] who extended the half-normal distribution and Iriarte et al.[6] who extended the Rayleigh distribution, calling it slash-Rayleigh (SR). A distribution which is very necessary in the paper is the gamma (Ga) distribution, whose probability density function (pdf) is given by
𝛽 𝛼 𝛼−1 −𝛽t t e , 𝛤 (𝛼)
g(t;𝛼, 𝛽) =
t > 0, 𝛼 > 0, 𝛽 > 0.
Its cumulative distribution function (cdf) is denoted by:
G(x;𝛼, 𝛽) =
∫0
x
g(t;𝛼, 𝛽)dt.
(2)
The Nakagami (NA) distribution (see Nakagami[12]) has been used for modeling fading-shadowi
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