Order-Constrained Inference for Multi-Level Step-Stress Accelerated Life Tests with the Gompertz Distribution

This work addresses statistical inference for multi-level step-stress accelerated life testing. Under the assumption that product lifetime follows the Gompertz distribution at each stress level, featuring a common shape parameter and stress-varying scale parameters, the cumulative exposure model (CEM) serves to link the lifetime distributions across different stress levels. Given the physical principle that lifetime decreases with an increase in stress, order-restricted parameter estimation methods have been developed. Within the classical frequentist framework, the order restriction is converted into box-constrained optimization via parameter reparameterization, obtaining maximum likelihood estimators (MLEs) and constructing asymptotic confidence intervals (CIs). With the Bayesian approach, weakly informative priors are adopted to prevent posterior impropriety, and a Markov chain Monte Carlo (MCMC) algorithm is employed for posterior sampling to avoid weight degeneracy in importance sampling. Extensive Monte Carlo simulations demonstrate that, after incorporating the order restriction, the Bayesian estimates yield smaller values in terms of bias, mean squared error (MSE), and interval length, while the MLEs provide classical asymptotic CIs as a reference. Lastly, a real dataset on fish swimming endurance is examined to demonstrate the practicality as well as the effectiveness of these proposed approaches.