Large sample properties of some survival estimators in heterogeneous samples

The Kaplan-Meier survival distribution estimator and Nelson hazard function estimator may not be appropriate descriptive devices for studies which involve nonhomogeneous populations. In this paper we investigate a mean survival estimator that incorporates important covariates associated with survival. The estimator is constructed based on Cox's regression model and a weighting according to any specified distribution of the baseline covariates. A mean hazard function estimator is also derived corresponding to the multiplicative intensity model which allows for recurrent outcome events. The weak convergence of each estimator is established using martingale and stochastic convergence results and its limiting variance is calculated. The strong and weak convergence of the Kaplan-Meier estimate and Nelson estimate are also explored in a nonhomogeneous population with an underlying proportional hazards model. It is shown that the Kaplan-Meier estimate remains a uniformly consistent estimate of S(t) = P(T > t) even in a heterogeneous sample if the censoring distribution is independent of covariates. However, when the censoring distribution depends on covariates, the Kaplan-Meier estimator can be quite biased even though the mean survival function estimator remains unbiased.