Likelihood and Bayesian Inference for Two Lomax Populations Under Balanced Joint Adaptive Progressive Type-II Censoring with an Exponential Ridge

Balanced joint adaptive progressive Type-II censoring (B-JAPC) is developed for two independent Lomax populations with a common shape parameter and population-specific scale parameters. Classical and Bayesian inference methods are constructed for the model parameters, survival functions, and hazard rates. To address the exponential scale–shape ridge where standard maximum likelihood estimates often fail to converge, a constrained maximum likelihood estimator (CMLE) with parametric bootstrap confidence intervals is established. A partially conjugate Bayesian framework under a Beta–Gamma prior is also implemented via a Metropolis-within-Gibbs algorithm. Monte Carlo simulations demonstrate that the proposed adaptive design substantially reduces the mean test duration compared to non-adaptive schemes while maintaining high inferential accuracy. The methodology is successfully applied to randomized cloud-seeding rainfall data, confirming its practical utility and quantifying the sensitivity of lifetime inference to shape regularization.