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.