Product-limit Estimators and Cox Regression with Missing Censoring Information

The Kaplan–Meier estimator of a survival function requires that the censoring indicator is always observed. A method of survival function estimation is developed when the censoring indicators are missing completely at random (MCAR). The resulting estimator is a smooth functional of the Nelson–Aalen estimators of certain cumulative transition intensities. The asymptotic properties of this estimator are derived. A simulation study shows that the proposed estimator has greater efficiency than competing MCAR-based estimators. The approach is extended to the Cox model setting for the estimation of a conditional survival function given a covariate.     

Tests for stochastic ordering under biased sampling

In two-sample comparison problems it is often of interest to examine whether one distribution function majorises the other, that is, for the presence of stochastic ordering. This paper develops a nonparametric test for stochastic ordering from size-biased data, allowing the pattern of the size bias to differ between the two samples. The test is formulated in terms of a maximally selected local empirical likelihood statistic. A Gaussian multiplier bootstrap is devised to calibrate the test. Simulation results show that the proposed test outperforms an analogous Wald-type test, and that it provides substantially greater power over ignoring the size bias. The approach is illustrated using data on blood alcohol concentration of drivers involved in car accidents, where the size bias is due to drunker drivers being more likely to be involved in accidents. Further, younger drivers tend to be more affected by alcohol, so in making comparisons with older drivers the analysis is adjusted for differences in the patterns of size bias.