Simultaneous Parametric Confidence Bands for Cumulative Distributions from Censored Data

This 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 provides a confidence region with coverage probabilities that are more accurate than a method based on bootstrapping the likelihood ratio statistic. We illustrate the implementation of the methods with an application to estimate probability of detection (POD), a function that is used to assess nondestructive evaluation (NDE) capability.