Exponentiated-based Burr type X distributions with censored data and covariate - Oh, Yit Leng (2024)
In an attempt to create distributions with greater flexibility to accommodate survival data with various hazard function forms, a number of extended Burr type X distributions have been developed recently. For instance, the Weibull Burr type X, beta Burr type X, and gamma Burr type X distributions. Previous studies have demonstrated that the hazard functions of these distributions can take various forms, such as increasing, decreasing, and bathtub, but not unimodal, which is frequently observed in survival analysis. In order to solve this issue, the aim of this study is to propose three distributions with greater flexibility in fitting hazard functions in various forms, particularly the unimodal: exponentiated Weibull Burr type X, exponentiated beta Burr type X, and exponentiated gamma Burr type X distributions. We begin by deriving the probability density function and cumulative distribution function of the three proposed distributions as well as their important statistical characteristics, including the quantile function, moment, moment generating function, order statistics, and Renyi entropy. To explore the performance of the three proposed distributions, simulation studies with various sample sizes and censoring rates for data with and without censored data and covariate are conducted after it. This study considers two types of censoring: random censoring and type-I censoring. Besides, for the simulation studies, we consider cases with single covariates. For these simulation studies, the inverse transform approach is used to simulate the event time. Meanwhile, we estimate the parameters of each of the three proposed distributions using the maximum likelihood estimation approach. Lastly, we use three real data sets: two complete data sets and one with censored data and covariates, to demonstrate the effectiveness and adaptability of the three suggested distributions. The two complete data sets are Data Set 1, which represents the failure time of 84 aircraft windshields, and Data Set 2, which represents the remission time of 128 patients with bladder cancer. Data Set 1 has an increasing hazard function, whereas Data Set 2 has a unimodal hazard function. Data Set 3 contains the recurrence time of 86 bladder cancer patients with censored data and three covariates. The findings of this study have demonstrated that the hazard functions of the three proposed distributions can take the forms of increasing, decreasing, bathtub, and unimodal. Additionally, even with censored data and covariate are present, the parameters of the three proposed distributions can be estimated using the maximum likelihood estimation approach. Finally, the three proposed distributions fit the two complete data sets better than various extended Burr type X distributions and their sub-models, and they are formidable rivals to all other competing distributions, including the non-nested distributions used in this study, as demonstrated in the real data applications. Nevertheless, the three proposed distributions can be used to fit survival data with unimodal hazard function as demonstrated in applications of Data Set 3.