Pak.j.stat.oper.res. Vol.17 No. 3 2021 pp 633-647 DOI: http://dx.doi.org/10.18187/pjsor.v17i3.3635 Assessing the Lifetime Performance Index of Burr Type III Distribution under Progressive Censoring 633 Assessing the Lifetime Performance Index of Burr Type III Distribution under Progressive Type II Censoring Amal S. Hassan 1 , Amany S. Selmy 2 , Salwa M. Assar 3* * Corresponding Author 1. Department of Mathematical Statistics, Faculty of Graduate Studies for Statistical Research , Cairo University, Egypt, dr.amalelmoslamy@gmail.com 2. Department of Mathematical Statistics, Faculty of Graduate Studies for Statistical Research, Cairo University, Egypt, amanyselmy34@gmail.com 3. Department of Mathematical Statistics, Faculty of Graduate Studies for Statistical Research, Cairo University, Egypt, salwaassar@yahoo.com Abstract Process capability analysis has been widely applied in the field of quality control to monitor the performance of industrial processes. Hence, lifetime performance index CL is used to measure the potential and performance of a process. In the present study, we construct a maximum likelihood estimator of CL under Burr type III distribution based on the progressive Type II censored sample. The maximum likelihood estimator of CL is then utilized to develop the hypothesis testing procedure in the condition of known L. Finally, one practical example and Monte Carlo simulation are given to assess the behavior of the lifetime performance index under given significance level. Key Words: Process capability index; Lifetime performance index; Burr Type III distribution, Progressive Type II censoring; Maximum likelihood estimator 1. Introduction Burr (1942) has suggested twelve types of cumulative distribution functions, which yield a variety of density shapes. The principal aim in choosing one of these forms of distributions is to facilitate the mathematical analysis to which it is subjected, while attaining a reasonable approximation. Burr Type III (BIII) distribution is a very important lifetime model in the analysis of equipment failure data, vehicle and informational studies, as well as in models of stress and durability. The lifetime X of product has a two-parameter BIII distribution if its cumulative distribution function (cdf) and probability distribution function (pdf) defined respectively as ( ) ; , 1 x , 0 , , 0, Fx x     − −   = +     (1) and, 1 1 (; , ; 0 ) . 1 fx x x x      − − − − −   = +    (2) where  and  are shape parameters. Altindag et al. (2017) proposed the estimation and prediction problems for the BIII distribution under type II censored data. Panahi (2017) developed the statistical inference of the unknown parameters of a BIII distribution based on the unified hybrid censored sample. The maximum likelihood (ML) estimates of the unknown parameters were obtained using the expectation–maximization algorithm. Gamchi et al. (2019) studied the estimation and prediction problems for the BIII distribution under progressive type II hybrid censored data. They obtained the ML estimates of unknown parameters using stochastic expectation maximization algorithm. Process capability indices (PCIs) have been used in the manufacturing industry to provide quantitative measures of process potential and performance. High quality production provides advantages such as cost saving, reduced scrap Pakistan Journal of Statistics and Operation Research