Estimation of the reliability parameter for three-parameter Weibull models

Abstract

• Proposal of correct likelihood for three-parameter Weibull models and inferences about their reliability parameter. • New insight for the relationship between the smallest observation and the threshold parameter. • Profile likelihood inferences for reliability parameter when Weibull models have common shape and threshold parameters. • Novel Bootstrap inferences for reliability parameter of threeparameter Weibull distributions with all parameters unknown. • Proposed likelihood can also be used with other densities having singularities and threshold parameter. For X and Y independent three-parameter Weibull random variables , the estimation of the reliability parameter δ = P ( Y < X ) is important in applications of failure times and stress-strength situations in industry, medicine, extreme value theory, hydrology, and environmetrics. Estimation problems arise when the likelihood function is not defined properly, since the three-parameter Weibull distribution is non regular and its density has singularities. Here, a correct likelihood for the three-parameter Weibull case is proposed for the first time, in the spirit of the original definition of likelihood. It takes into account the occurrence of the smallest observation with respect to the threshold parameter, as well as the fact that all measuring instruments necessarily have a finite precision. Possible repeated observations are immediately explained. When the Weibull distributions of X and Y have common threshold and shape parameters, the profile likelihood can be used for inferences about δ . For the case of all three Weibull parameters unknown and arbitrary, inferences about δ are obtained via a novel Bootstrap approach . An example previously analyzed under alternative inferential approaches is presented to illustrate the convenience of the proposal.