In recent years, for data with positive support, specifically, lifetime, or reliability, the half-normal (HN) model has been widely used. The probability density function (pdf) is given byf(x;σ)=2σϕ(xσ)I{x>0},where σ>0 is the scale parameter and ϕ(⋅) represents the standard normal pdf. We denote this by writing X∼HN(σ).Some generalizations for this model are proposed by Cooray and Ananda [1], Cordeiro et al. [2], Bolfarine and Gómez [3] and Gómez and Vidal [4].Olmos et al. [5] extended the HN distribution by incorporating a kurtosis parameter q, with the purpose of obtaining heavier tails, i.e., it has greater kurtosis than the base model. They called this model the slashed half-normal (SHN) distribution. Its construction is based on considering the quotient of two independent random variables, with random variable X∼HN(σ) in the numerator and the U∼U(0,1) in the denominator (See Rogers and Tukey [6] and Mosteller and Tukey [7] for more details). Thus a model is obtained that has more flexible coefficients of asymmetry and kurtosis than the HN model. We say that a random variable T follows a SHN if its pdf is given byfT(t;σ,q)=q2qπ−−−√σqΓ((q+1)/2)t−(q+1)G(t2;(q+1)/2,12σ2),t>0,(1)where σ>0 is a scale parameter, q>0 is a kurtosis parameter, G(z;a,b)=∫20g(x;a,b)dx is the cumulative distribution function (cdf) of the gamma distribution and g(⋅;a,b) is the pdf of the gamma model with shape and rate parameters a and b, respectively.Reyes et al. [8] introduced the modified slash (MS) distribution. We say that M has a MS distribution ifM=Z/E1q,(2)the construction of which is based on considering an exponential (Exp) distribution with parameter 2 in the denominator, i.e., they consider that E∼Exp(2). The motivation of the selection of the Exp(2) distribution is given in Reyes et al. [8]. The result of this work shows that the MS model has a greater coefficient of kurtosis and this characteristic is very important for modeling data sets when they contain atypical observations.The principal goal of this article is to use the idea published by Reyes et al. [8] to construct an extension of the half-normal model with a greater range in the coefficient of kurtosis than the SHN model, in order to use it to model atypical data. This will allow us obtain a new model generated on the basis of a scale mixture between an HN and a Weibull (Wei) distribution.The rest of the paper is organized as follows: Section 2 contains the representation of this model and we generate the density of the new family, its basic properties and moments, and its coefficients of asymmetry and kurtosis. In Section 3 we make inferences using the moments and maximum likelihood (ML) methods. In Section 4 we implement the expectation–maximization (EM) algorithm. In Section 5 we carry out a simulation study for parameter recovery. We show three illustrations in real datasets in Section 6 and finally in Section 7 we present our conclusions.
https://www.mdpi.com/2073-8994/11/9/1150/htm
Neveka M. Olmos, Osvaldo Venegas, Yolanda M. Gómez, Yuri A. Iriarte
