This article considers the estimation for bivariate distribution function (d.f.) <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$F_0(t, z)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>F</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> of survival time <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$T$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>T</mi> </math> </EquationSource> </InlineEquation> and covariate variable <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$Z$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>Z</mi> </math> </EquationSource> </InlineEquation> based on bivariate data where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$T$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>T</mi> </math> </EquationSource> </InlineEquation> is subject to right censoring. We derive the empirical likelihood-based bivariate nonparametric maximum likelihood estimator <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\hat{F}_n(t,z)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mover accent="true"> <mi>F</mi> <mo stretchy="false">^</mo> </mover> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$F_0(t,z)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>F</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation>, which has an explicit expression and is unique in the sense of empirical likelihood. Other nice features of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$\hat{F}_n(t,z)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mover accent="true"> <mi>F</mi> <mo stretchy="false">^</mo> </mover> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> include that it has only nonnegative probability masses, thus it is monotone in bivariate sense. We show that under <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$\hat{F}_n(t,z)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mover accent="true"> <mi>F</mi> <mo stretchy="false">^</mo> </mover> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation>, the conditional d.f. of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$T$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>T</mi> </math> </EquationSource> </InlineEquation> given <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$Z$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>Z</mi> </math> </EquationSource> </InlineEquation> is of the same form as the Kaplan–Meier estimator for the univariate case, and that the marginal d.f. <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$\hat{F}_n(\infty ,z)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mover accent="true"> <mi>F</mi> <mo stretchy="false">^</mo> </mover> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>∞</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> coincides with the empirical d.f. of the covariate sample. We also show that when there is no censoring, <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$$\hat{F}_n(t,z)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mover accent="true"> <mi>F</mi> <mo stretchy="false">^</mo> </mover> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> coincides with the bivariate empirical d.f. For discrete covariate <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$$Z$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>Z</mi> </math> </EquationSource> </InlineEquation>, the strong consistency and weak convergence of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$$\hat{F}_n(t,z)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mover accent="true"> <mi>F</mi> <mo stretchy="false">^</mo> </mover> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> are established. Some simulation results are presented. Copyright The Institute of Statistical Mathematics, Tokyo 2014