The jackknife’s edge: Inference for censored regression quantiles

For censored data, it is very common for the tail of the survival function to be non-identifiable because of the abundance of censored observations in the tail. This is especially prominent in censored regression quantile analysis, and introduces a serious problem with inference, especially near the point of non-identifiability. The lack of readily estimable formulas for asymptotic variances requires the use of resampling methods. Unfortunately, the bootstrap (in any of its versions) generates samples for which the point of non-identifiability has sufficient variability over the pseudo-samples so that (in theory and in practice) an appreciable number of the bootstrap replicates can no longer estimate a quantile that the original data could estimate. This leads to very poor coverage probabilities. Thus, resampling methods that provide less variability over the pseudo-samples may be very helpful. The jackknife is one such method, though it is necessary to use a “delete-d” jackknife with d of order larger than n. Another alternative is to use randomly reweighted “bootstrap” samples with weights of the form 1+v, with v of order 1/n. These approaches can be justified asymptotically. Furthermore, a simulation experiment shows that randomly sampling a relatively modest number of delete-(2n) jackknifed samples provides quite excellent coverage probabilities, substantially outperforming Bootstrap methods near the point of non-identifiability. This provides a counterexample to the commonly held notion that bootstrap methods are better than jackknife methods, and suggests the possible superiority of jackknife methods for a variety of situations with missing data or other cases of partial identifiability.