Reliability analysis of independent Burr-X competing risks model based on improved adaptive progressively Type-Ⅱ censored samples with applications

In the analysis of failure time data, researchers often encounter situations in which events arise from multiple mutually exclusive causes. This framework is commonly referred to as competing risks. Traditional methods are inadequate in such settings, as they typically assume the presence of a single type of failure and do not account for the influence of other competing events. This study investigates the competing risks model under an improved adaptive progressive Type-Ⅱ censoring scheme, which is particularly beneficial in contexts where the duration of testing is critical. The lifetimes associated with competing risks are assumed to follow independent Burr-X distributions, a flexible model capable of accommodating a variety of data types. Both classical and Bayesian estimation methods are utilized to estimate the model parameters and the reliability function, a key metric in reliability assessment. Maximum likelihood estimates are computed numerically, and approximate confidence intervals are derived. For Bayesian inference, squared error and linear-exponential loss functions are employed. Given the complexity of the posterior distribution, Markov chain Monte Carlo techniques are utilized to obtain Bayesian estimates and construct Bayesian credible intervals. A comprehensive numerical analysis, which includes a simulation study and the examination of real competing risk datasets, is conducted to evaluate and compare the performance of the proposed estimation methods.