Estimating the expectation of the log-likelihood with censored data for estimator selection

A criterion for choosing an estimator in a family of semi-parametric estimators from incomplete data is proposed. This criterion is the expected observed log-likelihood (ELL). Adapted versions of this criterion in case of censored data and in presence of explanatory variables are exhibited. We show that likelihood cross-validation (LCV) is an estimator of ELL and we exhibit three bootstrap estimators. A simulation study considering both families of kernel and penalized likelihood estimators of the hazard function (indexed on a smoothing parameter) demonstrates good results of LCV and a bootstrap estimator called ELL_bboot. We apply the ELL_bboot criterion to compare the kernel and penalized likelihood estimators to estimate the risk of developing dementia for women using data from a large cohort study.