Locally Efficient Estimation of the Quality Adjusted Lifetime Distribution with Right-Censored Data and Covariates

Zhao and Tsiatis (1997) consider the problem of estimation of the distribution of the quality adjusted lifetime when the chronological survival time is subject to right-censoring. The quality adjusted lifetime is typically defined as a weighted sum of the times spent in certain states until death or some other failure time. They propose an estimator and establish the relevant asymptotics, under the assumption of independent censoring. In this paper we extend the data structure with a covariate process observed until end of follow up and identify the optimal estimation problem. Because of the curse of dimensionality, no globally efficient nonparametric estimators with a good practical performance at moderate sample sizes exist. Given a correctly specified model for the hazard of censoring conditional on the observed state process, we propose a closed form one-step estimator of the distribution of the quality adjusted lifetime whose asymptotic variance attains the efficiency bound, if we can correctly specify a lower-dimensional working model for the conditional distribution of quality adjusted lifetime given the observed quality of life and covariate processes. The estimator remains consistent and asymptotically normal even if this latter submodel is misspecified. The practical performance of the estimators is illustrated with a simulation study. We also extend our proposed one-step estimator to the case where treatment assignment is confounded by observed risk factors so that it can be used to test a treatment effect in an observational study