Journal of Probability and Statistics

Volume 2018, Article ID 3152807, 12 pages

https://doi.org/10.1155/2018/3152807

## The Half-Logistic Lomax Distribution for Lifetime Modeling

Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad, Islamabad, Pakistan

Correspondence should be addressed to Masood Anwar; kp.ude.stasmoc@rawnadoosam

Received 7 August 2017; Revised 30 November 2017; Accepted 14 December 2017; Published 1 February 2018

Academic Editor: Chin-Shang Li

Copyright © 2018 Masood Anwar and Jawaria Zahoor. 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

We introduce a new two-parameter lifetime distribution called the half-logistic Lomax (HLL) distribution. The proposed distribution is obtained by compounding half-logistic and Lomax distributions. We derive some mathematical properties of the proposed distribution such as the survival and hazard rate function, quantile function, mode, median, moments and moment generating functions, mean deviations from mean and median, mean residual life function, order statistics, and entropies. The estimation of parameters is performed by maximum likelihood and the formulas for the elements of the Fisher information matrix are provided. A simulation study is run to assess the performance of maximum-likelihood estimators (MLEs). The flexibility and potentiality of the proposed model are illustrated by means of real and simulated data sets.

#### 1. Introduction

The commonly used lifetime distributions (exponential, gamma, Weibull, Lomax, lognormal, etc.) have a limited range of behavior and do not provide adequate fit to complex data sets in different sciences. Generalizations of these distributions offer more flexibility and provide reasonable parametric fits to complex data sets. Motivated by the various applications of Lomax and half-logistic distributions in areas of income and wealth inequality, firm size, size of cities, queuing problems, actuarial science, medical and biological sciences, and engineering, we propose a two-parameter continuous lifetime distribution by compounding the half-logistic and the Lomax distribution called* half-logistic Lomax (HLL) distribution*.

The Lomax [1] (or Pareto Type-II) distribution was introduced to model business failure data. For more detail about the Lomax distribution, we refer the readers to Rady et al. [2], Tahir et al. [3], and the references therein. In literature, there are several generalizations of the Lomax distribution. Abdul-Moniem [4] developed the exponentiated Lomax distribution, and Al-Awadhi and Ghitany [5] introduced the discrete Poisson–Lomax distribution by using the Lomax distribution as a mixing distribution for the Poisson parameter. Asgharzadeh et al. [6] proposed the Pareto Poisson–Lindely distribution, and Cordeiro et al. [7] investigated the gamma-Lomax distribution and studied its properties. Ghitany et al. [8] and Gupta et al. [9] considered the Marshal–Olkin approach and extended the Lomax distribution, and Lemonte and Cordeiro [10] proposed and studied the McDonald-Lomax, the beta Lomax, and the Kumaraswamy Lomax distributions. Other models constitute flexible family of distributions in terms of the variates of shapes and hazard functions; see, for example, Al-Zahrani and Sagor [11], El-Bassiouny et al. [12], Rady et al. [2], Kilany [13], and Tahir et al. [3]. These generalizations of the Lomax distribution are considered to be useful life distribution models.

The cumulative distribution function (cdf) of Lomax distribution is given by where is a shape parameter and is a scale parameter. The probability density function (pdf) corresponding to (1) is

Cordeiro et al. [14] define the cdf of the new* type I half-logistic-G *(TIHL-) family of distributions by where is the baseline cdf depending on a parameter vector and an additional shape parameter . As a special case, if , then the TIHL- is the half-logistic- (HL-) distribution with cdf

The corresponding pdf to (4) is given by

This paper aims to provide a new lifetime model with a minimum number of parameters by compounding the half-logistic and the Lomax distribution called half-logistic Lomax (HLL) distribution. The proposed distribution is heavy-tailed and has a decreasing or upside-down bathtub (or unimodal) shaped hazard rates depending on its parameters. Upside-down bathtub shaped hazard rates are common in reliability, engineering, and survival analysis. The HLL distribution can also be applied in engineering as the Lomax [1] distribution and can be a useful alternative to other well-known densities in lifetime applications. It is interesting to note that the HLL distribution is a special case of Marshall-Olkin–Lomax distribution introduced by Ghitany et al. [8]. We obtain some mathematical properties of the proposed distribution and parameters of the model are estimated by the maximum-likelihood estimation method.

The rest of this paper is organized as follows. In Section 2, we introduce the half-logistic Lomax distribution and provide plots of its density function. In Section 3, we investigate various mathematical properties of the HLL distribution including survival and hazard rate function, quantile function, moments, mean residual life function, mean deviation from the mean and the mean deviation from the median, entropies, and order statistics. In Section 4, estimation of parameters is given by MLE method and the asymptotic distribution of the estimators is studied via Fisher’s information matrix. Simulation results on the behavior of the MLEs are presented in Section 5. A real data application is conducted in Section 6. Finally, in Section 7, we conclude that the HLL distribution is the best model as compared to other competing models.

#### 2. The HLL Distribution

We define the* half-logistic-Lomax* (HLL) density function by inserting (1) and (2) into (5). So, we obtain The corresponding cumulative density function (cdf) follows from (1) and (4) and is given byHereafter, we will denote a random variable having pdf (6) by . The limit of the HLL density (6) as is 0 and the limit as is . Figure 1 depicts some of the possible shapes of density (6) for selected parameter values. The mode of density (6) is obtained from solving , which is given by