Nonparametric rank based estimation of bivariate densities given censored data conditional on marginal probabilities

In this note we develop a new Kaplan-Meier product-limit type estimator for the bivariate survival function given right censored data in one or both dimensions. Our derivation is based on extending the constrained maximum likelihood density based approach that is utilized in the univariate setting as an alternative strategy to the approach originally developed by Kaplan and Meier (1958). The key feature of our bivariate survival function is that the marginal survival functions correspond exactly to the Kaplan-Meier product limit estimators. This provides a level of consistency between the joint bivariate estimator and the marginal quantities as compared to other approaches. The approach we outline in this note may be extended to higher dimensions and different censoring mechanisms using the same techniques.