Parametric Simultaneous Confidence Bands for Cumulative Distributions From Censored Data

Abstract

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...

Simultaneous confidence bands and regions for log-location-scale distributions with censored data

“…As shown in the appendix of Jeng and Meeker (2001), for a given CR, there is equivalence between optimization procedures (5) and (6), which means (5) and (6) give the same SCB for the cdf. Cheng and Iles (1983) provided closed form solutions of (5) and (6) using expected information for complete data.…”

“…Cheng and Iles (1988) gave one-sided SCB s for a location-scale cdf with complete data. Jeng and Meeker (2001) compared coverage probabilities of SCB s based on a normal approximation, using observed information and expected information, likelihood, and bootstrap procedures. Their paper described the geometry of one versus two sided SCB s, and presented two examples.…”

“…Theorems 8 and 9 can be proved in a straightforward manner by using Lemma 4 in Appendix A.2. For more information about the coverage probability of these SCB s, see Jeng and Meeker (2001), who used extensive simulations to compare the coverage probability for different SCB s.…”

“…Usually, it is important to assess the precision of the cdf estimate. Jeng and Meeker (2001) present two example applications for single distribution local-scale models; one is on life data and the other one is on probability of detection in which the usual simple regression model is replaced by a physics-based computer model so that there is only an unknown location parameter and an unknown scale parameter. One approach of describing the uncertainty of the estimated cdf is to construct a simultaneous confidence band (SCB ) that contains the entire unknown cdf with a certain confidence level.…”

Coverage probabilities of simultaneous confidence bands and regions for log-location-scale distributions

“…The SCR for θ can be obtained from Wald statistics with expected information, estimated expected information, or local information (e.g., Escobar et al, 2009). It can also be obtained through inversion of a likelihood ratio or a score statistic or a parametric bootstrap procedure (e.g., Jeng and Meeker, 2001). Then one obtains the graph of all the cdfs F (y; θ) when θ is in the SCR(θ).…”