Comparisons of Approximate Confidence Interval Procedures for Type I Censored Data

Abstract

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.

A Comparison of Maximum Likelihood and Median-Rank Regression for Weibull Estimation

“…When a method has good statistical properties, one might expect that the associated interval method would also have good properties and vice versa. Of course one could and probably should study the properties of the interval method directly, as in Vander Weil and Meeker (1990) and Jeng and Meeker (2000).…”

“…We thus discarded any samples in which the number of failures was fewer than two, making our evaluation conditional on having at least two failures. Table 3 in Jeng and Meeker (2000) gives the number of observed samples in their simulation in which there were only 0 or 1 failures. Our results files contain similar information but these counts are not reported here, due to space constraints.…”

The step-stress tampered failure rate model under interval monitoring

“…The use of the BCa (adjusted percentile) bootstrap method (see Efron and Tibshirani [13]) is justified in Balakrishnan et al [8]. A discussion about bootstrap CIs for type-I censored data can also be found in Jeng and Meeker [20].…”

Avoiding Problems With Normal Approximation Confidence Intervals for Probabilities

“…Jeng and Meeker (2000) reported simulation results comparing the coverage probabilities of normal approximation, likelihood, and simulation-based CI procedures for estimating quantiles of the Weibull and lognormal distributions. Because there is a close relationship between CI procedures for a quantile and the z procedure for probabilities, the results of Jeng and Meeker (2000) provide insight into the relative behavior of coverage probability under type I censoring for CI procedures for probabilities.…”