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

Abstract

In many areas of application, especially life testing and reliability, it is often of interest to estimate an unknown cumulative distribution (cdf). A simultaneous confidence band (SCB) of the cdf can be used to assess the statistical uncertainty of the estimated cdf over the entire range of the distribution. Cheng and Iles (1983) presented a general approach of constructing an SCB for the cdf of a continuous random variable. For the loglocation-scale family of distributions, there are explicit forms for the upper and lower boundaries of the SCB. In this article, we extend the work of Cheng and Iles (1983). We study the SCBs based on local information, expected information, and estimated expected information for both the "cdf method" and the "quantile method. " We also study the effects of exceptional cases where a simple SCB does not exist. We describe calibration of the bands to provide exact coverage for complete data and type II censoring and better approximate coverage for other kinds of censoring. We also extend these procedures to regression analysis.

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

“…Using Corollary 1, it can be verified that the SCB procedures in Theorems 2 through 7 in Escobar et al (2009) have a CP exactly equal to (1 − α) if they are based on SCR procedures that have a CP exactly equal to (1 − α). Corollary 1 is useful in the calibration of an SCB procedure.…”

“…Thus, it suffices to calibrate the SCR procedure instead of calibrating the SCB procedure. See Escobar et al (2009) for more details on calibrating SCR procedures. …”

“…First, a 100(1 − α)% simultaneous confidence region (SCR), denoted by SCR(θ), is identified for the unknown parameters θ = (µ, σ ) of the cdf. 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).…”

“…Escobar et al (2009) extended the work of Cheng and Iles (1983) and studied the SCBs derived from local information, expected information, and estimated expected information for approaches (1) and (2), and considered special features of non-ellipsoidal SCRs that can occur when using the expected information.…”