Empirical likelihood bivariate nonparametric maximum likelihood estimator with right censored data

Abstract

This article considers the estimation for bivariate distribution function (d.f.) $$F_0(t, z)$$ of survival time $$T$$ and covariate variable $$Z$$ based on bivariate data where $$T$$ is subject to right censoring. We derive the empirical likelihood-based bivariate nonparametric maximum likelihood estimator $$\hat{F}_n(t,z)$$ for $$F_0(t,z)$$ , which has an explicit expression and is unique in the sense of empirical likelihood. Other nice features of $$\hat{F}_n(t,z)$$ include that it has only nonnegative probability masses, thus it is monotone in bivariate sense. We show that under $$\hat{F}_n(t,z)$$ , the conditional d.f. of $$T$$ given $$Z$$ is of the same form as the Kaplan–Meier estimator for the univariate case, and that the marginal d.f. $$\hat{F}_n(\infty ,z)$$ coincides with the empirical d.f. of the covariate sample. We also show that when there is no censoring, $$\hat{F}_n(t,z)$$ coincides with the bivariate empirical d.f. For discrete covariate $$Z$$ , the strong consistency and weak convergence of $$\hat{F}_n(t,z)$$ are established. Some simulation results are presented.