Estimation of the reliability parameter for three-parameter Weibull models

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.