Smoothing techniques for the bivariate kaplan-meier estimator

Bivariate survival time data arise quite often in medical research, and many estimators for the bivariate survival function have been suggested. While there are a lot of smooth estimators for the univariate Kaplan-Meier estimator, smooth versions of bivariate Kaplan-Meier estimator are not discussed yet. In this article, we suggest two smoothing techniques, the kernel smoothing and the Bezier surface smoothing, for the bivariate survival function estimator, especially for the estimator suggested by Lin and Ying (1993). Also, asymptotic results for both estimators are derived. Throughout the simulation studies, the Bezier surface smoothing turned out to be very efficient compared to the bivariate Kaplan-Meier estimator and the kernel smoothing estimator. An illustrative example based on a real data set is also given.