Hazard Estimation With Bivariate Survival Data and Copula Density Estimation

Abstract

Bivariate survival function can be expressed as the composition of marginal survival functions and a bivariate copula and, consequently, one may estimate bivariate hazard functions via marginal hazard estimation and copula density estimation. Leveraging on earlier developments on penalized likelihood density and hazard estimation, a nonparametric approach to bivariate hazard estimation is being explored in this article. The new ingredient here is the nonparametric estimation of copula density, a subject of interest by itself, and to accommodate survival data one needs to allow for censoring and truncation in the setting. A simple copularization process is implemented to convert density estimates into copula densities, and a cross-validation scheme is devised for density estimation under censoring and truncation. Empirical performances of the techniques are investigated through simulation studies, and potential applications are illustrated using real-data examples and open-source software.