Parameter estimation of Kumaraswamy Burr type X models based on cure models with or without covariates - Yusuf, Madaki Umar (2017)
In the last few years, many attempts have been made to define new models that extend well known Beta Kumaraswamy-G (BK-G) and Kumaraswamy-G (K-G) families of distribution that can provide a greater flexibility in modeling real-life data. Further-more, one of the weakness of Beta distribution is that it is not fairly tractable and in a particular case, its cumulative distribution function (CDF) involves the incomplete beta function ratio. Kumaraswamy distribution has a closed form of probability density function (PDF) and CDF, which makes it tractable. This motivated us to extend the BK-G family which has four shape parameters and K-G family which has two shape parameters. The Burr Type X (BX) distribution was chosen because of its PDF and CDF are of a closed form. Asa consequence of this, it can be used suitably for censored data. Based on the problem stated, we develop a new model using the method of confounding the existing parametric models by adopting the BX model as the baseline distribution. This proposed model is called Beta Kumaraswamy Burr-Type X (BKBX) distribution with six parameters. Due to the intricacy and non-close form solution of the BKBX model, we provide the modified better version of the model by reducing its parameters to four and called this as Kumaraswamy Burr-Type X (KBX) distribution. In this thesis, we considered two methods via the classical maximum likelihood estimation (MLE) and the Bayes estimation using the Gibbs sampling (G-S) algorithm to estimate the parameters of BKBX, KBX and Beta-Weibull (BWB) models. We obtained the posterior summaries considering the cure models with covariates by the method of Gibbs sampling of the Markov Chain Monte Carlo (MCMC). Series of simulation studies were conducted to evaluate the performance of the proposed estimation approaches. The two common types of cure fraction models, namely; mixture and the non-mixture models for the survival data based on the BKBX, KBX and BWB distributions were provided. Hence, to obtain effective results for the cure models with censored data and covariates, the estimation of the parameters was done under a Bayesian approach using G-S method. The comparison was done between the BKBX, KBX and BWB models to validate the usefulness of the modified distributions. The KBX signifies and proves to be less time consuming, have a close form solution of both its survival and hazard function unlike BKBX and BWB models and yet have similar features as the Kumaraswamy Weibull (KWB) distribution. Based on the results of the cure models, with or without covariates for the censored dataset used at all levels of comparison, the KBX model okto be the best choice. The application of real datasets which are uncensored and not cure model of right skewed, left skewed and approximate symmetry were considered. Based on the results obtained, the KBX distribution has provided a better fit compared to the BKBX, the baseline BX, and non-nested models based on the model selection criteria using the MLE.