Practical and Optimal Likelihood Intervals and Regions for Weibull and Gumbel Distributions

Profile likelihood intervals with good coverage probabilities and convenient simple expressions are presented to estimate separately and efficiently each of the parameters and quantiles of Weibull and Gumbel distributions, widely used in reliability and environmental applications. For the first time, likelihood-confidence intervals for these distributions, that were previously unavailable, can now be easily and quickly calculated with the provided R code for sample sizes n≥10. For simultaneous estimation of both parameters, likelihood-confidence regions are also provided. The proposal takes advantage of the transformation relationship between the Weibull and Gumbel distributions, of the fact that this latter distribution belongs to the Location and Scale family of distributions, and of likelihood symmetrizing parameterizations, all leading to simpler inferences about the parameters. The plot of the relative profile likelihood of a quantile of interest and its corresponding profile likelihood intervals provide decision makers with useful information in many practical applications, particularly when comparing two treatments. The proposed intervals for the Weibull shape parameter facilitate testing if an exponential distribution is reasonable for the data. The proposed methods are illustrated with practical examples.