Exact Inference on Weibull Parameters With Multiply Type-I Censored Data

Abstract

In reliability engineering, the Weibull distribution and censoring are widely employed. The multiple Type-I censoring is the general form of Type-I censoring and represents that all the test units are terminated at different times. In this paper, the exact inference on Weibull parameters under multiple Type-I censoring is discussed, as the existing studies are limited. The least-square estimates (LSE) and maximum likelihood estimates (MLE) of Weibull parameters are presented. For the problem that the MLE does not always exist and has no closed-form, the approximate maximum likelihood estimate (AMLE) is proposed. Concerning the confidence interval (CI) for Weibull parameters, the pivotal quantity based on the LSE is developed to construct it. Another CI is obtained using the asymptotic normality of MLE. By the missing information principle, the observed Fisher information matrix is derived to replace the commonly approximate one through evaluating the second order derivatives of log-likelihood function with the MLE. As a comparison, the CI via the bootstrap method is provided. Monte Carlo simulation demonstrates that the proposed AMLE is convenient and superior to the MLE. And the proposed CI based on pivotal quantity outperforms others. Besides, the derived observed Fisher information matrix is also better. Finally, a real example is analyzed to illustrate the application of proposed methods.