# Optimal Confidence Regions for Weibull Parameters and Quantiles under Progressive Censoring
###### Abstract  
Confidence regions for the Weibull parameters with minimum areas among all those based on the Conditionality Principle are constructed using an equivalent diffuse Bayesian approach. The process is valid for scenarios involving standard failure and progressive censorship, and complete data. Optimal conditional confidence sets for two Weibull quantiles are also derived. Simulation-based algorithms are provided for computing the smallest-area regions with fixed confidence levels. Importantly, the proposed confidence sets satisfy the Sufficiency, Likelihood and Conditionality Principles in contrast to the unconditional regions based on maximum likelihood estimators and other insufficient statistics. The suggested perspective can be applied to parametric estimation and hypothesis testing, as well as to the determination of minimum-size confidence sets for other invariantly estimable functions of the Weibull parameters. A dataset concerning failure times of an insulating fluid is studied for illustrative and comparative purposes. confidence and credibility regions; Conditionality, Likelihood and Sufficiency Principles; Weibull distribution; progressively censored data; reliability and life-testing +
Footnote †: journal: 2023 by the authors  
The development of confidence sets for Weibull parameters and quantiles from test samples is of great interest and practical relevance in many experimental analyses. In particular, these regions are useful for model selection, parametric estimation and hypothesis testing. In practice, joint confidence sets for the Weibull scale and shape parameters, \(\theta\) and \(\alpha\), are often based on the unconditional distribution of pivotal quantities related to the maximum likelihood estimator (MLE) of \((\theta,\alpha)\), denoted as \((\widehat{\theta},\widehat{\alpha})\). However, \((\widehat{\theta},\widehat{\alpha})\) does not contain all the sample information. In the Weibull case, the whole sample is minimally sufficient. Confidence regions for \((\theta,\alpha)\) can also be derived by using the conditional distribution of \((\widehat{\theta},\widehat{\alpha})\), given the observed values of the ancillary statistics. From a strictly logical point of view, the conditional approach seems more appropriate because it obeys the Sufficient Principle. Uniformly most accurate confidence sets do not exist in the Weibull case. Region size minimization is an alternative optimality criterion that is commonly used to select the best confidence set; see, for example, [PERSON] and [PERSON] [30], [PERSON] and [PERSON] [31] and [PERSON] [32]. At first glance, as smaller confidence sets contain fewer points, they are less likely to cover false values. Interval estimation from progressively censored data has been considered by many authors, including [PERSON] and [PERSON] [33], [PERSON] [34], [PERSON] [35] and [PERSON] [36]. This paper deals with the construction of minimum-area confidence regions for Weibull parameters and quantiles based on progressively censored data when the principle of conditioning on ancillary statistics is adopted. Our approach is based on the general results of [PERSON] and [PERSON] [37] for location-scale parameter problems, which are also valid for log-location-scale models, such as the Weibull distribution, because the logarithmic transformation is strictly monotonic. According to [PERSON] and [PERSON] [37], conditional frequentist confidence sets and Bayesian credibility sets for invariantly estimable functions are numerically equivalent when the analyst assumes independent Lebesgue measure priors on the location parameter and the logarithm of the scale parameter. In our setting, the Weibull parameters and quantiles correspond to invariantly estimable functions, and the proposed optimal confidence regions would include the points with the highest posterior density (HPD) assuming independent flat priors for \(\log(\theta)\) and \(\log(\alpha)\).