On Algorithms and Approximations for Progressively Type‐I Censoring Schemes

ABSTRACT This research aims to study the estimation of parameters for a Burr‐XII distribution and to investigate optimal sampling plans under progressive Type‐I censoring (PTIC). For point estimation, we employed the maximum‐likelihood estimation (MLE) method using two numerical approaches: Newton–Raphson and the Expectation–Maximization algorithm. We also utilized Bayesian estimation with squared error loss and linear exponential loss functions. Specifically, two approximate Bayesian methods, the Lindley and Tierney‐Kadane methods were examined. Additionally, Bayesian numerical estimation was performed using Markov Chain Monte Carlo with the Metropolis‐Hastings (MH) algorithm. For interval estimation, we constructed asymptotic confidence intervals for MLE and the highest posterior density method within the Bayesian framework. The practical study involved Monte Carlo simulations to assess the efficiency and accuracy of the proposed estimation methods across different PTIC schemes. A real data analysis is also provided to illustrate the practical application of these methodologies in analyzing a clinical trial dataset. Data from a clinical trial using a PTIC scheme reveals patterns in pain relief, aiding in evaluating the antibiotic ointment's effectiveness. The study further investigates optimal sampling plans for the Burr‐XII distribution under PTIC.