Comparisons of Approximate Con dence Interval Procedures for Type I Censored Data

This paper compares di erent procedures to compute con dence intervals for parameters and quantiles of the Weibull, lognormal, and similar log-location-scale distributions from Type I censored data that typically arise from life test experiments. The procedures can be classi ed into three groups. The rst group contains procedures based on the commonlyused normal approximation for the distribution of studentized (possibly after a transformation) maximum likelihood estimators. The second group contains procedures based on the likelihood ratio statistic and its modi cations. The procedures in the third group use a parametric bootstrap approach, including the use of bootstrap-type simulation, to calibrate the procedures in the rst two groups. The procedures in all three groups are justi ed on the basis of large-sample asymptotic theory. We use Monte Carlo simulation to investigate the nite sample properties of these procedures. Details are reported for the Weibull distribution. Our results show, as predicted by asymptotic theory, that the coverage probabilities of one-sided con dence bounds calculated from procedures in the rst and second groups are further away from nominal than those of two-sided con dence intervals. The commonly-used normal-approximation procedures are crude unless the expected number of failures is large (more than 50 or 100). The likelihood ratio procedures work much better and provide adequate procedures down to 30 or 20 failures. By using bootstrap procedures with caution, the coverage probability is close to nominal when the expected number of failures is as small as 15 to 10 or less, depending on the particular situation. Exceptional cases, caused by discreteness from Type I censoring, are noted.