Performance Index for Weibull Distribution With the Progressive First-Failure-Censored Sampling

Process capability indices are essential tools for assessing product quality, offering insights into how well a product meets specified standards and customer expectations. This paper explores the lifetime performance index under the assumption that product lifetimes follow a Weibull distribution. We first develop six generalized approaches using the concept of a generalized pivotal quantity and incorporate properties of progressively first-failure-censored samples, as well as maximum likelihood and weighted least squares estimation for the Weibull distribution. Additionally, we introduce a Bayesian approach for inference on the lifetime performance parameter and employ Markov chain Monte Carlo simulations to compute posterior estimates and construct credible intervals. Furthermore, we extend another Bayesian approach that uses a different strategy for specifying the prior distribution of parameters—a transition from progressive Type II censoring to progressive first-failure censoring. We analyze the properties of confidence intervals for the performance index through Monte Carlo simulations, comparing the results of a large-sample method based on maximum likelihood estimation and the extended Bayesian approach with the seven novel approaches. The behavior of the confidence intervals for the parameter is illustrated through two examples.