Locally Efficient Estimation with Current Status Data and Time-Dependent Covariates

In biostatistical applications, interest often focuses on the estimation of the distribution of a failure time variable T. If one observes only whether or not T exceeds an observed monitoring time C, then the data structure is called current status data, also known as interval-censored data, case I. We extend the data structure by allowing the presence of a possibly time-dependent covariate process which is observed up till the monitoring time C. We follow the approach of Robins and Rotnitzky by modeling the hazard of C conditional on the failure time variable and the covariate-process (i.e., the missingness or censoring process) under the restriction that the missingness (monitoring) process satisfies coarsening at random. Because of the curse of dimensionality, no globally efficient nonparametric estimators with a good practical performance at moderate sample sizes exist. We introduce an inverse probability of censoring weighted estimator of the distribution and of smooth functionals of this distribution that are guaranteed to be consistent and asymptotically normal if we have available a correctly specified parametric or semiparametric model for the missingness process. Furthermore, given a correctly specified model for the missingness process, we propose a locally efficient one-step estimator whose asymptotic variance attains the efficiency bound, if we correctly specify a lower-dimensional model for the conditional distribution of T given the covariates. The estimator remains consistent and asymptotically normal even if this latter submodel is misspecified. We conclude with a simulation experiment and a data analysis.