Inference on the reliability of Weibull distribution with multiply Type-I censored data

• We focus on reliability of Weibull distribution under multiply Type-I censoring.
• The proposed confidence interval for the reliability is superior after comparison.
• The Bayes estimates with a few expert judgements on reliability are satisfactory.
• We specify the cases where the MLEs do not exist and present methods to remedy it.
• The distribution of estimate of reliability should be used for accurate estimate.

In this paper, we focus on the reliability of Weibull distribution under multiply Type-I censoring, which is a general form of Type-I censoring. In multiply Type-I censoring in this study, all units in the life testing experiment are terminated at different times. Reliability estimation with the maximum likelihood estimate of Weibull parameters is conducted. With the delta method and Fisher information, we propose a confidence interval for reliability and compare it with the bias-corrected and accelerated bootstrap confidence interval. Furthermore, a scenario involving a few expert judgments of reliability is considered. A method is developed to generate extended estimations of reliability according to the original judgments and transform them to estimations of Weibull parameters. With Bayes theory and the Monte Carlo Markov Chain method, a posterior sample is obtained to compute the Bayes estimate and credible interval for reliability. Monte Carlo simulation demonstrates that the proposed confidence interval outperforms the bootstrap one. The Bayes estimate and credible interval for reliability are both satisfactory. Finally, a real example is analyzed to illustrate the application of the proposed methods.