Empirical likelihood ratio confidence intervals for conditional survival probabilities with right censored data

In the analysis of survival data, we often encounter situations where the response variable (the survival time) T is subject to right censoring, but the covariates Z are completely observable and are often discrete or categorical. In this article, we construct the empirical likelihood ratio confidence region for conditional survival probabilities based on bivariate data which are subject to right censoring in one coordinate and have a discrete covariate Z. We show that such an empirical likelihood ratio confidence region is indeed an interval, and we establish some related properties of the empirical likelihood ratio. The generalization of our results in this article to the multivariate covariate Z with dimension p > 1 is straightforward.