On Expectation Measures for Failure Processes in Multiple Populations: Mathematical Theory and Applications on Two Lines

This paper develops classical and Bayesian inferential procedures for Weibull exponential lifetime models under joint progressive Type-II censoring, motivated by comparative reliability analysis of products manufactured across multiple production lines. The theoretical framework is formulated for a general setting involving k independentWeibull exponential populations, allowing for flexible modeling of heterogeneous lifetime behaviors under a common censoring scheme. Maximum likelihood estimators and their asymptotic confidence intervals are derived, and Bayesian estimation is conducted using Markov chain Monte Carlo methods under both squared-error and LINEX loss functions. For numerical illustration and practical interpretability, the primary emphasis of the simulation study, expected-failure analysis, and real-data applications is placed on the two-population case (k = 2), which commonly arises in comparative life-testing scenarios such as the evaluation of two production lines or systems. Explicit expressions for the expected number of failures are presented for two populations, and their performance is examined through Monte Carlo simulations under various censoring schemes. The proposed methods are further illustrated using real datasets, demonstrating their applicability and effectiveness in reliability assessment. Overall, the results show that the proposed inferential procedures perform well under joint progressive censoring and provide a useful statistical framework for comparative reliability analysis, with methodology that naturally extends to general k-population settings.