Journal of Probability and Statistics

Volume 2018, Article ID 8767826, 12 pages

https://doi.org/10.1155/2018/8767826

## The Half-Logistic Generalized Weibull Distribution

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 8 August 2017; Revised 17 November 2017; Accepted 28 November 2017; Published 4 January 2018

Academic Editor: Ahmed Z. Afify

Copyright © 2018 Masood Anwar and Amna Bibi. 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

A new three-parameter generalized distribution, namely, half-logistic generalized Weibull (HLGW) distribution, is proposed. The proposed distribution exhibits increasing, decreasing, bathtub-shaped, unimodal, and decreasing-increasing-decreasing hazard rates. The distribution is a compound distribution of type I half-logistic-G and Dimitrakopoulou distribution. The new model includes half-logistic Weibull distribution, half-logistic exponential distribution, and half-logistic Nadarajah-Haghighi distribution as submodels. Some distributional properties of the new model are investigated which include the density function shapes and the failure rate function, raw moments, moment generating function, order statistics, L-moments, and quantile function. The parameters involved in the model are estimated using the method of maximum likelihood estimation. The asymptotic distribution of the estimators is also investigated via Fisher’s information matrix. The likelihood ratio (LR) test is used to compare the HLGW distribution with its submodels. Some applications of the proposed distribution using real data sets are included to examine the usefulness of the distribution.

#### 1. Introduction

Statistical distributions are the basic aspects of all parametric statistical techniques including inference, modeling, survival analysis, and reliability. For the analysis of lifetime data, it is an important task to fit the data by a statistical model. A number of lifetime distributions have been developed in literature for this purpose. The widely used lifetime models usually have a limited range of behaviors. Such type of distributions cannot give a better fit to model all the practical situations. Recently, several authors have developed a number of families of statistical models by applying different techniques. Various techniques have been introduced in the literature to derive new flexible models as discussed by Lai [1].

Marshall and Olkin [2] introduced an effective technique to extend a family of distributions by addition of another parameter. By applying this technique, they generalized the exponential and the Weibull distributions. Al-Zahrani and Sagor [3] proposed the Poisson Lomax model by compounding the Poisson Lomax distributions. Bidram and Nadarajah [4] introduced the exponentiated EG2 distribution by using the method of resilience parameter. Kus [5] considered compounding of Poisson and exponential distribution. Gurvich et al. [6] generalized the Weibull distribution offering a new distribution function elucidating a wide range of functional forms of the effect of size on the strength distribution, using a simple method of evaluation of the basic statistical parameters. Nadarajah and Kotz [7], Lai et al. [8], Lai et al. [9], and Xie et al. [10] further discussed some modifications of the Weibull model.

In this paper, another extension of the extended Weibull distribution is introduced using half-logistic-G generator. So we propose the half-logistic generalized Weibull (HLGW) distribution without adding any extra parameter to the baseline model. The new model is the compound distribution of two previously known distributions, one of which follows the class proposed by Gurvich et al. [6] and the other is type I half-logistic-G model. The proposed model is able to depict more complex hazard rates and provides a good alternate to the Weibull distribution that does not exhibit upside-down bathtub-shaped or unimodal failure rates.

Dimitrakopoulou et al. [11] established a three-parameter lifetime model with PDFwhere and are the shape parameters and is a scale parameter. The CDF corresponding to (1) isIn this paper, a three-parameter lifetime model is presented which is the compound model of the previously known models introduced by Dimitrakopoulou et al. [11] and half-logistic-G (HL-G) distribution called half-logistic generalized Weibull (HLGW) distribution. The half-logistic-G distribution is presented by Cordeiro et al. [12] with the CDF where is the CDF of the baseline distribution and is the shape parameter. As a special case, for , the TIHL-G is the half-logistic-G (HL-G) model with cumulative distribution functionThe corresponding PDF to (4) is given bywhere and .

The rest of the paper is unfolded as follows. Section 2 contains the introduction of the half-logistic generalized Weibull (HLGW) distribution and provides the plots of its density function. Section 3 explores the distributional properties of the HLGW model. In Section 4, the method of maximum likelihood estimation is used to obtain the estimators of unknown parameters. The asymptotic distribution of the estimators is also investigated in this section via Fisher’s information matrix. A simulation study is discussed in Section 5 to check out the accuracy of point and interval estimates of the HLGW parameters. Section 6 involves some applications of the HLGW distribution using lifetime data sets to examine the fitness of the proposed model. Section 7 provides concluding remarks about the paper.

#### 2. The Half-Logistic Generalized Weibull Distribution

Substitution of (1) and (2) in (5) results the following PDF of the HLGW distribution:The CDF associated with (6) is as follows:The parameters are the shape parameters and is a scale parameter. From now on, a random variable with PDF (6) will be written as ~.

#### 3. Distributional Properties

This section deals with the investigation of the distributional properties of HLGW distribution. The statistical properties include the plots of the density function, the failure rate function, raw moments, moment generating function, order statistics, L-moments, and quantile function.

##### 3.1. Special Cases

The HLGW distribution includes the following distributions as special cases:(i)For , the HLGW model reduces to the half-logistic Weibull (HLW) model with the PDF where is the shape parameter and is the scale parameter. For , the half-logistic Weibull model is also called the power half-logistic distribution (PHLD) proposed and studied by Krishnarani [13].(ii)For , the HLGW model generates a new model, the half-logistic exponential (HLE) model, with scale parameter and the PDF (iii)For , the HLGW model gives another new model, the half-logistic Nadarajah-Haghighi (HLNH) model, with the PDF where and are the shape and scale parameters, respectively.

##### 3.2. The Shapes of HLGW Distribution

The following are the conditions under which the PDF of the HLGW distribution (6) shows different behaviors:(i)For , the PDF is monotone decreasing with , .(ii)For , the same shape is exhibited with and .(iii)For , the PDF has the value zero at the origin; then it increases to a higher value and then decreases, falling towards the value of zero at infinity.

Various behaviors of the PDF are shown in Figure 1, for some parameter values; . The mode of density (6) can be obtained from .