Nonparametric Estimation of the Bivariate Survivor Function

Abstract

Nonparametric estimators of the bivariate survivor function have the potential to provide a basic tool for the display and comparison of survival curves, analogous to the Kaplan-Meier estimator for univariate failure time data. Available nonparametric estimators include estimators that plug empirical estimators of single and double failure hazard rates into survivor function representations, and versions of nonparametric maximum likelihood estimators (NPMLE) that address uniqueness problems. In this paper we consider candidate bivariate survivor function estimators that arise either from representations of the survivor function in terms of the marginal survivor functions and double failure hazard, or in terms of the double failure hazard only for a suitably truncated version of the data. The former class of estimators includes the Dabrowska (1988) and Prentice-Cai (1992) estimators, for which a marginal hazard-double failure hazard representation leads to suggestions for several new estimators. The estimators in this class tend to incorporate substantial negative mass, but corresponding proper estimators can be obtained by defining a restricted estimator that is either equal to the unrestricted estimator, or is as close as possible to the unrestricted estimator without violating non-negativity constraints. The second class of estimators includes an estimator arising from the simple empirical double failure hazard, as well more efficient estimators that redistribute singly censored observations within the strips of a partition of the risk region, following Van der Laan’s (1996) repaired NPMLE, as well as related adaptive estimators. These and selected other estimators are compared in simulation studies, leading to a synthesis of available estimation techniques, and to suggestions for future research.