Bayesian Estimation of Gumbel Type-II Distribution under Type-II Censoring with Medical Applications

The time to event or survival time usually follows certain skewed probability distributions. These distributions encounter vital role using the Bayesian framework to analyze and project the maximum life expectancy in order to inform decision-making. The Bayesian method provides a flexible framework for monitoring the randomized clinical trials to update what is already known using prior information about specific phenomena under uncertainty. Additionally, medical practitioners can use the Bayesian estimators to measure the probability of time until tumor recurrence, time until cardiovascular death, and time until AIDS for HIV patients by considering the prior information. However, in clinical trials and medical studies, censoring is present when an exact event occurrence time is not known. The present study aims to estimate the parameters of Gumbel type-II distribution based on the type-II censored data using the Bayesian framework. The Bayesian estimators cannot be obtained in explicit forms, and therefore we use Lindley's approximation based on noninformative prior and various loss functions such as squared error loss function, general entropy loss function, and LINEX (linear exponential) loss function. The maximum likelihood and Bayesian estimators are compared in terms of mean squared error by using the simulation study. Furthermore, two data sets about remission times (in months) of bladder cancer patients and survival times in weeks of 61 patients with inoperable adenocarcinoma of the lung are analyzed for illustration purposes.