Journal of Probability and Statistics

Volume 2017 (2017), Article ID 6303462, 9 pages

https://doi.org/10.1155/2017/6303462

## Modified Slash Lindley Distribution

^{1}Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta, Chile^{2}Departamento de Ciencias Matemáticas y Físicas, Facultad de Ingeniería, Universidad Católica de Temuco, Temuco, Chile

Correspondence should be addressed to Osvaldo Venegas

Received 27 September 2016; Revised 17 January 2017; Accepted 29 January 2017; Published 19 February 2017

Academic Editor: Ramón M. Rodríguez-Dagnino

Copyright © 2017 Jimmy Reyes 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 paper we introduce a new distribution, called the modified slash Lindley distribution, which can be seen as an extension of the Lindley distribution. We show that this new distribution provides more flexibility in terms of kurtosis and skewness than the Lindley distribution. We derive moments and some basic properties for the new distribution. Moment estimators and maximum likelihood estimators are calculated using numerical procedures. We carry out a simulation study for the maximum likelihood estimators. A fit of the proposed model indicates good performance when compared with other less flexible models.

#### 1. Introduction

The Lindley distribution was introduced by Lindley [1] with the density function given byWe denote this by writing , where is the shape parameter. The corresponding cumulative distribution function (c.d.f.) isThe properties of the Lindley distribution are studied in detail by Ghitany et al. [2]. Jodrá [3] uses the Lambert function for the generation of random variables with Lindley or Poisson-Lindley distributions. Ghitany et al. [4] propose the power Lindley (PL) distribution generated from a random variable raised to the power ; that is, if then has PL distribution of parameters and , with density function given byWe denote this by writing . The half-normal distribution is suitable for fitting positive data. We say that a random variable follows a half-normal distribution with scale parameter if its density function is given bywhere represents the density function of the standard normal distribution; we denote this by writing .

Olmos et al. [5] introduce a new distribution suitable for fitting positive data called the slash half-normal distribution, which is a distribution with right-tails heavier than the HN distribution. It is obtained as a particular case when the shape parameter tends to infinity. We say that a random variable follows a slash half-normal (SHN) distribution with scale parameter and kurtosis parameter if its density function is given byWe denote this by writing .

In the study of symmetric distributions with heavy tails, Reyes et al. [6] introduce a modification of the class of standard slash distributions, which will be called modified slash (MS) distribution and is described as follows: We will say that has MS distribution with parameter if it can be expressed aswhere and for . Here and are independent random variables, denoted by ; the density function of has heavier tails than the standard slash distribution and in consequence has higher kurtosis. When we obtain the standard normal distribution. The density function of the variable is given bywhere is kurtosis parameter; see Reyes et al. [6] for more details. Using the same idea, Reyes et al. [7] extend the skew-normal model and Reyes et al. [8] extend the Birnbaum-Saunders model. Gui [9] introduces the slash Lindley (SL) distribution and applies it to data on precipitation and plasma ferritin concentration. Gui et al. [10] introduce the Lindley-Poisson (LP) distribution and apply it to lifetime data.

The focus of this paper is the introduction of a new distribution called modified slash Lindley (MSL) distribution. Because of its mixed approach, the newly constructed distribution will have heavier tails than its parent Lindley distribution and hence will be more suitable for modeling positive data sets that may have heavy tails and/or outliers. This new distribution is quite appropriate for modelling positive data with very atypical observations (outliers); as we can see in the application, the MSL distribution MSL better models data on the survival cancer patients with atypical remission times.

The paper is organized as follows. Section 2 is devoted to the development of a stochastic representation for the MSL distribution and its use for density function derivation and also the derivation of its moments, asymmetry and kurtosis coefficients. In Section 3, the inference is discussed for the MSL distribution using the method of moments estimators and maximum likelihood estimation. We also present an illustrative example with real data on survival times. This example shows that the proposed distribution is a very appropriate model for this data set.

#### 2. MSL Distribution

In this section we consider a stochastic representation, the density function (with some graphical representations), and properties of the modified slash Lindley distribution.

##### 2.1. Stochastic Representation

The stochastic representation of the new distribution is given aswhere and are independent random variables with , . We called the distribution of the MSL distribution, and we use the notation .

##### 2.2. Density Function

The following result shows that the density function of the random variable MSL can be generated using the stochastic representation in (8).

Proposition 1. *Let . Then, the density function of is given bywith and .*

*Proof. *Using the stochastic representation in (8) and from the Jacobian transformation approach it follows thatHence,By marginalizing, the result follows immediately.

The following proposition shows that the MSL distribution results from a mixture of the LI distribution on the scale parameter and shape parameter , and the exponential distribution with parameter equals two.

Proposition 2. *Let and ; then .*

*Proof. *We can write

In Figure 1, we illustrate the behavior of the density function of the MSL distribution.