This article compares different procedures to compute confidence 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 classified into three groups. The first group contains procedures based on the commonly used 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 modifications. The procedures in the third group use a parametric bootstrap approach, including the use of bootstrap-type simulation, to calibrate the procedures in the first two groups. The procedures in all three groups are justified on the basis of large-sample asymptotic theory. We use Monte Carlo simulation to investigate the finite-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 confidence bounds calculated from procedures in the first and second groups are further away from nominal than those of two-sided confidence 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.

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This article describes existing methods and develops new methods for constructing simultaneous confidence bands for a cumulative distribution function. Our results are built on extensions of previous work by Cheng and Iles for two-sided and one-sided bands, respectively. Cheng and Iles used Wald statistics with (expected) Fisher information. We consider three alternatives-Wald statistics with observed Fisher information, Wald statistics with local information, and likelihood ratio statistics. We compare standard large-sample approximate methods with simulation or bootstrap-calibrated versions of the same methods. For (log-)location-scale distributions with complete or failure (Type II) censoring, the bootstrap methods have the correct coverage probability. A simulation for the Weibull distribution and time-censored (Type I) data shows that bootstrap methods provide coverage probabilities that are closer to nominal than those based on the usual large-sample approximations. We illustrate the methods with examples from product-life analysis and nondestructive evaluation probability of detection. KeywordsBootstrap, Life data, Likelihood ratio, Probability of detection, Reliability, Simultaneous confidence band, Wald Disciplines Statistics and Probability CommentsThis preprint was published in Technometrics 43 (2001) AbstractThis paper describes existing methods and develops new methods for constructing simultaneous confidence bands for a cumulative distribution function (cdf). Our results are built on extensions of previous work by Cheng and Iles (1983, 1988). Cheng and Iles use Wald statistics with (expected) Fisher information and provide different approaches to find one-sided and two-sided simultaneous confidence bands. We consider three statistics, Wald statistics with Fisher information, Wald statistics with local information, and likelihood ratio statistics. Unlike pointwise confidence intervals, it is not possible to combine two 95% one-sided simultaneous confidence bands to get a 90% two-sided simultaneous confidence band. We present a general approach for construction of two-sided simultaneous confidence bands on a cdf for a continuous parametric model from complete and censored data. Both two-sided and one-sided simultaneous confidence bands for the location-scale parameter model are discussed in detail including situations with complete and censored data.We start by using standard large-sample approximations and then extend and compare these to corresponding simulation or bootstrap calibrated versions of the same methods. The results show that bootstrap methods provide more accurate coverage probabilities than those based on the usual large sample approximations. A simulation for the Weibull distribution and Type I censored data is used to compare finite sample properties. For the location-scale model with complete or Type II censoring, the bootstrap methods are exact. Simulation results show that, with Type I censoring, a bootstrap method based on the Wald statistic with local information provid...

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