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In making inference on the relation between failure and exposure histories inthe Cox semiparametric model, the maximum partial likelihood estimator (MPLE)of the finite dimensional odds parameter, and the Breslow estimator of thebaseline survival function, are known to achieve full efficiency when data isavailable for all time on all cohort members, even when the covariates are timedependent. When cohort sizes become too large for the collection of completedata, sampling schemes such as nested case control sampling must be used and,under various models, there exist estimators based on the same information asthe MPLE having smaller asymptotic variance. Though the MPLE is therefore notefficient under sampling in general, it approaches efficiency in highlystratified situations, or instances where the covariate values are increasinglyless dependent upon the past, when the covariate distribution, not depending onthe real parameter of interest, is unknown and there is no censoring. Inparticular, in such situations, when using the nested case control samplingdesign, both the MPLE and the Breslow estimator of the baseline survivalfunction achieve the information lower bound both in the distributional and theminimax senses in the limit as the number of cohort members tends to infinity.

Efficiency of the maximum partial likelihood estimator for nested case control sampling