This article considers the estimation for bivariate distribution function (d.f.) F0(t,z)\documentclass[12pt]{minimal}
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\begin{document}$$F_0(t, z)$$\end{document} of survival time T\documentclass[12pt]{minimal}
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\begin{document}$$T$$\end{document} and covariate variable Z\documentclass[12pt]{minimal}
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\begin{document}$$Z$$\end{document} based on bivariate data where T\documentclass[12pt]{minimal}
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\begin{document}$$T$$\end{document} is subject to right censoring. We derive the empirical likelihood-based bivariate nonparametric maximum likelihood estimator F^n(t,z)\documentclass[12pt]{minimal}
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\begin{document}$$\hat{F}_n(t,z)$$\end{document} for F0(t,z)\documentclass[12pt]{minimal}
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\begin{document}$$F_0(t,z)$$\end{document}, which has an explicit expression and is unique in the sense of empirical likelihood. Other nice features of F^n(t,z)\documentclass[12pt]{minimal}
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\begin{document}$$\hat{F}_n(t,z)$$\end{document} include that it has only nonnegative probability masses, thus it is monotone in bivariate sense. We show that under F^n(t,z)\documentclass[12pt]{minimal}
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\begin{document}$$\hat{F}_n(t,z)$$\end{document}, the conditional d.f. of T\documentclass[12pt]{minimal}
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\begin{document}$$T$$\end{document} given Z\documentclass[12pt]{minimal}
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\begin{document}$$Z$$\end{document} is of the same form as the Kaplan–Meier estimator for the univariate case, and that the marginal d.f. F^n(∞,z)\documentclass[12pt]{minimal}
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\begin{document}$$\hat{F}_n(\infty ,z)$$\end{document} coincides with the empirical d.f. of the covariate sample. We also show that when there is no censoring, F^n(t,z)\documentclass[12pt]{minimal}
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\begin{document}$$\hat{F}_n(t,z)$$\end{document} coincides with the bivariate empirical d.f. For discrete covariate Z\documentclass[12pt]{minimal}
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\begin{document}$$Z$$\end{document}, the strong consistency and weak convergence of F^n(t,z)\documentclass[12pt]{minimal}
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\begin{document}$$\hat{F}_n(t,z)$$\end{document} are established. Some simulation results are presented.
