Frequentist and Bayesian Predictive Inference for the Log-Logistic Distribution Under Progressive Type-II Censoring

Area of Science:

• Statistics
• Survival Analysis
• Reliability Engineering

Background:

• Accurate prediction of future failure times is crucial in reliability and survival analysis.
• Heavy-tailed distributions, like the Log-Logistic, present unique challenges in failure time prediction.
• Progressive Type-II censoring is a common data collection scheme in reliability studies.

Purpose of the Study:

• To develop and compare frequentist and Bayesian methods for parameter estimation and future failure time prediction.
• To evaluate the performance of different point predictors (BUP, CMP, BP) and interval predictors (frequentist, ETI, HPD).
• To assess the reliability of these methods for the Log-Logistic distribution under progressive censoring.

Main Methods:

• Maximum Likelihood Estimation (MLE) and Bayesian estimation for model parameters.
• Derivation of Best Unbiased Predictor (BUP), Conditional Median Predictor (CMP), and Bayesian Predictor (BP).
• Construction of frequentist, Equal-Tailed Interval (ETI), and Highest Posterior Density (HPD) prediction intervals.
• Implementation of Bayesian methods using Markov Chain Monte Carlo (MCMC) sampling.
• Validation through Monte Carlo simulations and analysis of real-world datasets (bladder cancer, guinea pig survival).

Main Results:

• The Bayesian Predictor (BP), especially with an empirical prior, demonstrated the most accurate and stable point prediction performance.
• Frequentist prediction methods showed reduced reliability in heavy-tailed scenarios.
• The Bayesian Highest Posterior Density (HPD) intervals consistently outperformed other methods for interval prediction.
• Bayesian HPD intervals effectively reduced interval lengths for right-skewed data while maintaining coverage probability.

Conclusions:

• The Bayesian approach, particularly the BP and HPD intervals, is recommended for predicting future failures from a Log-Logistic distribution under progressive censoring.
• The study highlights the limitations of frequentist methods in extreme heavy-tailed settings.
• The findings offer practical guidance for reliability engineers and statisticians dealing with censored survival data.