Journal of Probability and Statistics

Volume 2018, Article ID 9753439, 10 pages

https://doi.org/10.1155/2018/9753439

## A New Class of Distributions Generated by the Extended Bimodal-Normal Distribution

Departamento de Matemática, Facultad de Ingeniería, Universidad de Atacama, Copiapó, Chile

Correspondence should be addressed to Juan F. Olivares-Pacheco; lc.adu@ravilofj

Received 21 June 2018; Accepted 27 September 2018; Published 1 November 2018

Academic Editor: Steve Su

Copyright © 2018 Milton A. Cortés et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this study, we present a new family of distributions through generalization of the extended bimodal-normal distribution. This family includes several special cases, like the normal, Birnbaum-Saunders, Student’s , and Laplace distribution, that are developed and defined using stochastic representation. The theoretical properties are derived, and easily implemented Monte Carlo simulation schemes are presented. An inferential study is performed for the Laplace distribution. We end with an illustration of two real data sets.

#### 1. Introduction

Although the normal distribution is the most popular probability model in statistics, several random phenomena in nature cannot be described by the normal distribution. In this regard, Azzalini [1] introduced an extension of the normal distribution called skew-normal distribution, where this model shares some properties with the standard normal model; it is mathematically tractable and it has a wide range of the coefficients of skewness and kurtosis. From this work, an important line of research focusing on finding new distributions that offered greater flexibility is generated.

More recently, Elal-Olivero [2] introduced a new class of skew-normal distribution called alpha-skew-normal distribution. In doing so, he first defined a new bimodal-symmetric normal distribution with probability density function given by where is the standard normal density, which is defined as the bimodal-normal (BN) distribution. Furthermore, he studies some properties of this distribution and presents its stochastic representation as the product of two independent random variables and , where and is a discrete random variable such that ; that is, has the distribution BN. On the other hand, an extension of the BN density is given bywhere is the shape parameter. Note that this density function is symmetric and is characterized by incorporating bimodality into the normal distribution, which is controlled by the parameter . Elal-Olivero [2] presents this extension as the symmetric-component of the alpha-skew-normal distribution. Furthermore, (2) also can be deduced from the model presented in Elal-Olivero et al. [3]. In this regard, Gui et al. [4] incorporated (2) into the slash distribution, developed its properties, and performed inferential studies, whereas Gómez and Guerrero [5] incorporated (2) into the Birnbaum-Saunders distribution, tested its bimodality, and demonstrated its principal properties.

The objective of this article is to present a new family of distributions through generalization of (2). This generalization can be applied to any density function, thereby producing a more flexible model incorporating a shape parameter. Depending on the density at which we apply this generalization, it is observed that the new model is flexible enough to support uni- and bimodal shapes. Furthermore, Gui et al. [4] and Gómez and Guerrero [5] are particular cases of the generalization proposal.

This article is organized as follows. In Section 2, we present a generalization of (2) and review some particular cases (normal, Birnbaum-Saunders, Student’s , and Laplace distribution). In Section 3, we develop the basic properties of the cases from Section 2 and study the effects of this new generalization. In Section 4, we study some inferential aspects of the extended Laplace distribution using maximum likelihood estimation and perform a Monte Carlo simulation study. We conclude in Section 5 with a discussion.

#### 2. A General Class of Distributions

This section describes a general class of distributions generated by (2), presents its basic properties, and derives explicit expressions for the normal, Birnbaum-Saunders, Student’s , and Laplace distribution.

##### 2.1. Characterization and Properties

Theorem 1 (general class of distributions). *Let be a probability density function and a positive continuous function such that , where . Then,is a probability density function with shape parameter .*

*Proof. *If we note can be represented as a mixture of two densities, then the result follows immediately; that is, , where .

*Remark 2. *On the basis of Theorem 1, we can make the following observations: (1)If , then , .(2)If , then , .

Theorem 3 (stochastic representation). *Let and be independent random variables. Ifthen .*

*Proof. *Since can be represented as a mixture, the result follows immediately.

*Remark 4. *If , then (1)The cumulative distribution function is given by where and are the cumulative distribution functions of and , respectively.(2)The moment generating function is given by where and are the moments generating functions of and , respectively, if both exist.(3)The -th moment of the random variable is given by

##### 2.2. Special Cases

In this section, explicit expressions are provided for the probability density function in (3) for the normal, Birnbaum-Saunders, Student’s , and Laplace distribution and different choices of . These models are selected to show the benefits of the proposed extension, and the choice of the function is conditioned upon a positive function with finite expectation.

Corollary 5 (normal case). *If and , then has the probability density function given byand we say that has an “extended normal distribution,” which is denoted as .*

Corollary 6 (Birnbaum-Saunders case). *Let . If and where is the derivative with respect to , with and , then has the probability density function given byand we say that has an “extended Birnbaum-Saunders distribution,” which is denoted as .*

Corollary 7 (Student’s case). *If and where , then has the probability density function given byand we say that has an “extended Student’s distribution,” which is denoted as .*

Corollary 8 (Laplace case). *If and , then has the probability density function given byand we say that has an “extended Laplace distribution,” which is denoted as .*

As we notice, in the Corollaries 5–8 and Figure 1, when the function is a symmetric density, the effect of the extension is that the model supports uni- and bimodal shapes. On the other hand, if the model has positive support, the bimodality depends on the choice of parameters, as seen in the Birnbaum-Saunders distribution case.