Parameters Estimation and Reliability Analysis for Burr XII Distribution Under Adaptive Progressive First-Failure Censoring: Systematic Techniques with Application

An adaptive progressive first-failure censoring scheme is used to enhance the efficiency of statistical analyses and minimize test time in life-testing experiments. This paper focuses on statistical inferences for the unknown parameters, survival, and hazard rate functions of the Burr XII distribution under this censoring scheme. Since the maximum likelihood estimates for the model parameters and reliability characteristics cannot be obtained explicitly, the Newton–Raphson method is employed for numerical derivation. The delta method is used to determine the variances of reliability characteristics and is applied to construct confidence intervals. Bayesian estimates of the unknown parameters and reliability characteristics are derived under the squared error and linear exponential loss functions. As these estimates are not explicitly obtainable, the Lindley and Markov chain Monte Carlo methods are used as approximation techniques. Additionally, asymptotic confidence intervals and highest posterior density credible intervals are developed for the parameters and reliability characteristics. A Monte Carlo simulation is performed to evaluate the proposed estimators, and the methodology is validated through a real dataset analysis on arthritic patients.