Journal of Probability and Statistics

Volume 2019, Article ID 8024769, 13 pages

https://doi.org/10.1155/2019/8024769

## On the Alpha Power Transformed Power Lindley Distribution

^{1}Institute of Statistical Studies and Research, Cairo University, Egypt^{2}Vice Presidency for Graduate Studies and Scientific Research, University of Jeddah, Jeddah, Saudi Arabia^{3}Statistics Department, Faculty of Science, King AbdulAziz University, Jeddah, Saudi Arabia

Correspondence should be addressed to M. Elgarhy; moc.oohay@58yhragle_m

Received 26 September 2018; Accepted 13 November 2018; Published 1 January 2019

Academic Editor: Ramón M. Rodríguez-Dagnino

Copyright © 2019 Amal S. Hassan 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 generalization of the power Lindley distribution referred to as* the alpha power transformed power Lindley* (APTPL). The APTPL model provides a better fit than the power Lindley distribution. It includes the alpha power transformed Lindley, power Lindley, Lindley, and gamma as special submodels. Various properties of the APTPL distribution including moments, incomplete moments, quantiles, entropy, and stochastic ordering are obtained. Maximum likelihood, maximum products of spacings, and ordinary and weighted least squares methods of estimation are utilized to obtain the estimators of the population parameters. Extensive numerical simulation is performed to examine and compare the performance of different estimates. Two important data sets are employed to show how the proposed model works in practice.

#### 1. Introduction

The one-parameter Lindley distribution was primarily proposed by Lindley [1] as an alternative model for data with a nonmonotone hazard rate shape. Ghitany et al. [2] provided the properties of the Lindley distribution under a careful mathematical treatment. They also displayed in a numerical application that the Lindley distribution provides a better model than the exponential distribution. A random variable* Y* is said to follow the Lindley distribution with scale parameter *θ* > 0, if its* probability density function* (pdf) is given by which is two-component mixture of exponential () and gamma (). The Lindley distribution has only increasing failure rate which has been identified as a major difficulty in lifetime analysis. To overcome this situation, many generalizations of the Lindley distribution have been introduced in literature. Nadarajah et al. [3] proposed a generalized Lindley distribution. Bakouch et al. [4] introduced the* extended Lindley* (EL) distribution. Shanker et al. [5] introduced the two-parameter Lindley distribution. Ghitany et al. [6] proposed the* power Lindley* (PL) as an extension of Lindley distribution. Based on pdf (1) and by using the transformation* X* = , the pdf of the PL distribution is given byThe* cumulative distribution function* (cdf) corresponding to (2) is given by Many other generalizations of Lindley distribution have been proposed by several authors; see for example, a new generalized Lindley distribution [7], transmuted quasi Lindley distribution [8], exponentiated power Lindley [9], transmuted generalized Lindley distribution [10], transmuted generalized quasi Lindley distribution [11], and transmuted Kumaraswamy quasi Lindley distribution [12].

The* alpha power transformation* (APT) is one of the procedures that make the distributions richer and flexible to model the real life data. The APT has been proposed by Mahdavi and Kundu [13] with the parameter to incorporate skewness to the base distribution. The cdf of APT is specified byMahdavi and Kundu [13] applied the APT method to the exponential distribution and they studied various properties of the alpha power exponential distribution. Based on APT method, some new distributions are considered like APT Weibull [14], APT generalized exponential [15], APT Lindley [16], APT extended exponential [17], and APT inverse Lindley [18].

Following the similar idea of Mahdavi and Kundu [13], we introduce a new three-parameter distribution, the so-called APTPL distribution. The main purpose of the new model is that the additional parameter can give several desirable properties and more flexibility in the form of the hazard and density functions. The APTPL distribution contains a number of known lifetime submodels such as the APT* Lindley* (APTL) distribution, PL distribution, Lindley distribution, and gamma distribution. Properties, estimation of the parameters and applications with real data, are considered.

This paper is constructed as follows. We introduce and study the properties of APTPL in Sections 2 and 3, respectively. In Section 4, we obtain the* maximum likelihood* (ML) estimator,* least square* (LS) estimator,* weighted least squares* (WLS) estimator, and* maximum product of spacing* (MPS) estimator of APTPL distribution. In the same section, a simulation study is executed to investigate the effectiveness of the estimates. The analyses of two real data sets are employed in Section 5. The paper ends with concluding remarks.

#### 2. The APTPL Model

Here, we now introduce the notion of APTPL distribution.

*Definition 1. *A random variable* X *is said to have a three-parameter APTPL distribution with the scale parameter *θ*>0 and shape parameters *α*,* β >*0, if its cdf is of the formwhere is a set of parameters. The pdf of the APTPL distribution is given byAlso, the reliability function, say and

*hazard rate function*(hrf), say of

*X*are given, respectively, as follows:andA random variable

*X*with distribution (6) is denoted by

*X*~ APTPL ( Submodels of the APTPL distribution are as follows:(i)For

*β*=1, the pdf (6) reduces to APTL [16].(ii)For =1, the pdf (6) reduces to PL [6].(iii)For

*β*=1 and =1, the pdf (6) reduces to Lindley [1].(iv)For

*β*=1 and =1, the pdf (6) reduces to gamma (2,

*θ*). Some descriptive pdf and hrf plots of

*X*are illustrated Figures 1 and 2 for specific parameter choices of (see Figures 1 and 2).