Parameter estimations and copula methods for burr type III and type XII distributions

Abstract

Continuous Burr distributions have gained popularity recently due to their potential use in practical situations. In particular, Burr Type III and Type XII distributions are suitable to describe lifetime data since these distributions, not only have flexible shape but also controllable scale and location parameters which are needed in characterizing lifetime distributions. In this study, 2-parameter and 3-parameter Burr Type III and XII distributions are employed to fit a set of simulated lifetime data. These lifetime data are assumed to be either complete, that is uncensored, or censored at varying levels of censoring, and are simulated from the specified Burr distributions using their inverse cumulative distribution functions. The distribution parameters are then estimated by using the classical maximum likelihood estimation (MLE) and expectation-maximization (EM) algorithm approaches. The performance of parameter estimates are then compared in terms of their accuracy and efficiency by comparing its bias and mean square errors. The study finds that as the censoring level varies, the EM estimates perform better than the MLE estimates for 2-parameter and 3-parameter Burr Type III and XII distributions with complete and censored lifetime data at certain censoring levels. In addition, the study also investigates a number of copula methods to join specific Burr Type III and XII distributions. The result reveals that Ali-Mikhail-Haq, Clayton and Gumbel methods fit well with Burr distributions for uncensored lifetime data since the values of copula lie within (0,1) interval.