Statistical Inference for Alpha Power Burr-XII Distribution Under Progressive Type-II Censoring: Methodology and Application to Bladder Cancer Data

In this paper we investigates the statistical inference and parameter estimation for the three-parameter Alpha Power Burr-XII (APB-XII) distribution under a Progressive Type-II censoring scheme. Both classical and Bayesian estimation methodologies are developed to estimate the unknown shape parameters (α,β, λ) as well as the corresponding reliability and hazard rate functions. Under the classical framework, maximum likelihood estimators (MLEs) are derived, and because they lack analytical solutions, the Newton-Raphson numerical technique is implemented. Furthermore, approximate asymptotic confidence intervals are constructed using the inverse Fisher information matrix alongside the delta method, complemented by parametric percentile (Boot-p) and studentized (Boot-t) bootstrap confidence intervals to address small sample limitations. Under the Bayesian framework, independent gamma prior distributions are utilized, and Bayes estimators are obtained under both symmetric Squared Error Loss (SEL) and asymmetric Linear Exponential (LINEX) loss functions. Due to the analytical complexity of the joint posterior distribution, the Markov Chain Monte Carlo (MCMC) algorithm specifically utilizing the Gibbs sampler within a Metropolis-Hastings approach is utilized to generate posterior samples and compute credible intervals. To demonstrate the real-world utility of the proposed methodology, a real lifetime dataset is modeled. The Kolmogorov-Smirnov test validates that the APB-XII distribution provides a highly robust fit for the empirical failure data. Finally, an extensive Monte Carlo simulation study is conducted under three distinct progressive censoring schemes.