Bivariate Current Status Data

In many applications, it is often of interest to estimate a bivariate distribution of two survival random variables. Complete observation of such random variables is often incomplete. If one only observes whether or not each of the individual survival times exceeds a common observed monitoring time C, then the data structure is referred to as bivariate current status data (Wang and Ding, 2000). For such data, we show that the identifiable part of the joint distribution is represented by three univariate cumulative distribution functions, namely the two marginal cumulative distribution functions, and the bivariate cumulative distribution function evaluated on the diagonal. The EM algorithm can be used to compute the full nonparametric maximum likelihood estimator of these three univariate cumulative distribution functions; however, we show that smooth functionals of these univariate cumulative cdfs can be efficiently estimated with easy to compute nonparametric maximum likelihood estimators (NPMLE), based on reduced data consisting of univariate current status observations. We use these univariate current status NPMLEs to obtain both a test of independence of the two survival random variables, and a test of goodness of fit for the copula model used in Wang & Ding (2000). Finally, we extend the data structure by allowing the presence of covariates, possibly time-dependent processes that are observed until the monitoring time C. We show that applying the locally efficient estimator, developed in van der Laan and Robins (1998), to the reduced univariate current status data yields locally efficient estimators