Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 7035279, 12 pages

http://dx.doi.org/10.1155/2016/7035279

## A Mixture of Two Burr Type III Distributions: Identifiability and Estimation under Type II Censoring

Department of Mathematics, College of Science, Aljouf University, P.O. Box 848, Sakaka 42421, Saudi Arabia

Received 16 June 2016; Revised 18 September 2016; Accepted 3 October 2016

Academic Editor: Juan C. Agüero

Copyright © 2016 A. S. Al-Moisheer. 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

The mixture of two Burr Type III distributions (MTBIIID) is investigated. First, the identifiability property of the MTBIIID is proved. Then, two different methods of estimation are used. Next, the estimates of the unknown five parameters and reliability function of the MTBIIID under Type II censoring are obtained. To study the performance of the estimation technique in the paper, a Monte Carlo simulation is presented. In addition, the numerical illustration requires solving nonlinear equations; therefore, the software international mathematical statistical library (IMSL) is used to assess these effects numerically. Finally, a real data set is applied to illustrate the methods proposed here.

#### 1. Introduction

Finite mixtures of distributions have been used as models throughout the history of statistics. For more details and examples about finite mixture of distributions see Everitt and Hand [1], Titterington et al. [2], McLachlan and Basford [3], Lindsay [4], McLachlan and Krishnan [5], and McLachlan and Peel [6]. Identifiability property is an important problem in mixture models since it gives a unique representation for a class of mixtures. Identifiability of mixtures has been discussed by several authors. Among others, they were discussed by Ahmad [7], Sultan et al. [8], and Sultan and Al-Moisheer [9].

The MTBIIID has its probability density function (pdf) as where , , , and , the density function of the th component, is given by

The cumulative distribution function (cdf) and reliability function of the MTBIIID, respectively, are given by where and , the cdf and reliability function of the th component, are given by

Recently, Al-Moisheer and Sultan [10] have proposed new mixture of two Burr Type III distributions (MTBIIID) with common shape parameter. They have proved that the MTBIIID with common shape parameter is identifiable. Also, they have derived the nonlinear discriminant function of the MTBIIID and calculated the total probabilities of misclassification as well as the percentage bias. Al-Moisheer [11] has investigated the problem of updating discriminant functions estimated from the MTBIIID.

Researches dealing with the method of maximum likelihood estimation of finite mixture models have been studied, for example, by Sultan et al. [8] and Sultan and Al-Moisheer [9]. The basic idea of Lindley’s [12] approximation for estimation under Type II censoring of the parameters of the mixture model has been investigated by many authors; among others, see Ahmad et al. [13] for a mixture of two Weibull distributions and Sultan and Al-Moisheer [14–16] in the case of mixture of two inverse Weibull distributions. A computationally simpler approximation is given by Tierney and Kadane [17], which requires the unimodality of the posterior distribution; such condition cannot be guaranteed in the case of MTBIIID. So, this approach will not be considered in this paper.

The main purpose of this paper is to prove that the MTBIIID with unknown shape parameters is identifiable in Section 2. The maximum likelihood estimators of the parameters and reliability function under Type II censoring are obtained in Section 3. In Section 4, Lindley’s procedure is applied to approximate the Bayes estimation of the unknown parameters and the reliability function based on Type II censoring. In Section 5, a simulation study is carried out to illustrate the estimation techniques considered in Sections 3 and 4. In addition, the usefulness of the proposed model is illustrated by fitting it to a new area of applications such as carbon monoxide level. Finally, in Section 6, some concluding remarks are drawn.

#### 2. Identifiability

Chandra [18] has proved the following.

Let be a transform associated with each having the domain of definition with linear map . If there exists a total ordering of such that(i), implies ,(ii)for each , there exists some , such that for , ,then class of all finite mixing distributions is identifiable relative to .

By using Chandra’s approach, we prove the following proposition.

Proposition 1. *The class of all finite mixing distributions relative to the Burr III distribution (BIIID) is identifiable.*

*Proof. *Let be a random variable having the pdf and cdf of the BIIID given in (2) and (5), respectively. Then the th moments of the th BIII component is given by where is the standard beta function.

From (5), we have Now let , , , and ; then from (8), we have that and see Abramowitz and Stegun [19].

On the other hand, when and , we have From (9) and (10), we have and hence the identifiability is proved.

#### 3. Maximum Likelihood Estimation

Suppose that is a censored sample under Type II right censoring of size obtained from a life test on items whose life times have a MTBIIID with density (1); the likelihood function takes the form where and , , are given, respectively, by (2) and (6). The log likelihood function in this case is then given by Equation (13) can be reduced to the complete sample case by setting . Differentiating with respect to 5-dimensional vector of parameters and setting it to zero, we obtain the following system of nonlinear equations: where, for and , and, for , , , , and are given in (1)–(6), respectively.

The solution of nonlinear equations (14) gives the MLEs of the vector of the five parameters. The routine DNEQNF from IMSL manual is used to solve this system of equations. The MLE of the reliability function is given by (4) after replacing the parameters by their MLEs. Table 1 displays the MLEs of the unknown parameters and the corresponding reliability function. Note that the MLEs of the unknown parameters of MTBIIID with common shape parameter based on the complete sample case are obtained by Al-Moisheer and Sultan [10].