Unified Confidence Bounds for Censored Weibull Data With Covariates

Data from a 2-parameter Weibull distribution (with covariates) is traditionally analyzed on a logarithmic scale to take advantage of the resulting location-scale nature of the transformed data Y1, . . . , Yn. Quantities of interest are the regression parameters β, the scale parameter σ, the p-quantile yp(u) = u′β+σ log(− log(1− p)), the tail probability p(y|u) = P (Y ≤ y|u) = 1−exp(− exp((y−u′β)/σ))), and the failure rate function r(y,u) = [exp((y − u′β)/σ)]/σ for a given p-dimensional covariate vector u. Usually such data is partially obscured by some sort of censoring which (aside from the case of type II censoring) does not allow exact confidence bounds for these quantities. Thus one resorts to large sample approximations from maximum likelihood theory. Unfortunately this has led to different types of approximations depending on the quantity of interest, see Meeker and Escobar (1998). For example, confidence bounds for yp(u) and p(y|u) are not always monotone in p or y and thus are not constructed as inverses to each other as would be the case when exact methods are possible. Also, one usually invokes approximate normality of the m.l.e.’s ŷp(u) and log(σ̂) (the latter for producing better results) with the apparent inconsistency that ŷp(0) = σ̂ log(− log(1 − p)) invokes the inferior approximation for σ̂. We resolve these problems by invoking either the approximate (p+ 1)-variate normal approximation for ((β̂ − β)/σ̂, log(σ̂)) or its bootstrapped approximating distribution. This resolves all the above problems in a clean fashion and in the bootstrap case it leads back to the approach by Robinson (1983).