Nonparametric estimation of univariate and bivariate survival functions under right censoring: a survey

Survival analysis studies time to event data, also called survival data in biomedical research. The main challenge in the analysis of survival data is to develop inferential methods that take into account the incomplete information contained in censored observations. The seminal paper of Kaplan and Meier (J Am Stat Assoc 53:457–481,1958) gave a boost to the development of statistical methods for time to event data subject to right censoring; methods that have been applied in a broad variety of scientific fields including health, engineering and economy. A basic quantity in survival analysis is the survival distribution: $$S(t) = P(T > t)$$ S ( t ) = P ( T > t ) , with T the time to event or, in case of a bivariate vector of lifetimes $$(T_1,T_2)$$ ( T 1 , T 2 ) , $$S(t_1,t_2) = P(T_1> t_1,T_2 > t_2$$ S ( t 1 , t 2 ) = P ( T 1 > t 1 , T 2 > t 2 ). Nonparametric estimation of these basic quantities received, since Kaplan and Meier (J Am Stat Assoc 53:457–481,1958), considerable attention resulting in many publications scattered over a large period of time and a large field of applications. The purpose of this paper is to review, in a unified way, nonparametric estimation of S(t) and $$S(t_1,t_2)$$ S ( t 1 , t 2 ) for time to event data subject to right censoring. Interesting to realize is that, in the multivariate setting, the form of the nonparametric estimator for $$S(t_1,t_2)$$ S ( t 1 , t 2 ) is determined by the actual censoring scheme. In this survey we focus, for the proposed (implicitly) existing or new nonparametric estimators, on the asymptotic normality. By doing so we fill some gaps in the literature by introducing some new estimators and by providing explicit expressions for the asymptotic variances often not yet available for some of the existing estimators.