Bayesian Estimations of Exponential Distribution Based on Interval-Censored Data with a Cure Fraction

Censored data are considered to be of the interval type where the upper and lower bounds of an event's failure time cannot be directly observed but only determined between interval inspection times. The analyses of interval-censored data have attracted attention because they are common in the fields of reliability and medicine. A proportion of patients enrolled in clinical trials can sometimes be cured. In some instances, their symptoms mostly disappear without any recurrence of the disease. In this study, the proportion of such patients who are cured is estimated. Furthermore, the Bayesian approach under the gamma prior and maximum likelihood estimation (MLE) is used to estimate the cure fraction depending on the bounded cumulative hazard (BCH) model based on interval-censored data with an exponential distribution. The Bayesian approach uses three loss functions: squared error, linear exponential, and general entropy. These functions are compared with the MLE and used between estimators. Moreover, they are obtained using the mean squared error, which locates the best option to estimate the parameter of an exponential distribution. The results show that the BCH model and lambda parameter of the exponential distribution based on the interval-censored data can be best estimated using the Bayesian gamma prior with a positive loss function of the linear exponential.