Order restricted inference for Chen life time populations under progressive type-II censoring scheme with partially observed competing risks

This paper examines the estimation of a competing risks model when failure reasons are partially observed. Specifically, we explore the scenario in which the Chen distribution governs the latent lifetime data of competing risk units under a progressive type-II censoring scheme. Estimations are conducted using both classical and Bayesian approaches, taking into account non-order cases and order restrictions on scale parameters under both unknown and known shape parameters.The uniqueness and existence of maximum likelihood estimates for the undetermined parameters are discussed. Subsequently, asymptotic confidence intervals and modified confidence intervals are calculated from the observed Fisher matrix. Additionally, we perform Bayesian inference for the model parameters utilizing a balanced loss function, subsequently deriving the credible interval. Furthermore, we examine the conditions under which supplementary order information is available for the competing risk parameters, and under these circumstances, we derive both classical and Bayesian estimates. Monte Carlo simulations are used to evaluate the effectiveness of the proposed estimation methods. Furthermore, we assess two real datasets to demonstrate the practical utility of these methods. Finally, we investigate the optimal censoring scheme by conducting tests based on various criteria.