In survival analysis, the relationship between a survival time and a covariate is conveniently modeled with the proportional hazards regression model. This model usually assumes that the covariate has a log-linear effect on the hazard function. In this paper we consider the proportional hazards regression model with a nonparametric risk effect. We discuss estimation of the risk function and its derivatives in two cases: when the baseline hazard function is parametrized and when it is not parametrized. In the case of a parametric baseline hazard function, inference is based on a local version of the likelihood function, while in the case of a nonparametric baseline hazard, we use a local version of the partial likelihood. This results in maximum local likelihood estimators and maximum local partial likelihood estimators, respectively. We establish the asymptotic normality of the estimators. It turns out that both methods have the same asymptotic bias and variance in a common situation, even though the local likelihood method uses information about the baseline hazard function.
"Local likelihood and local partial likelihood in hazard regression." Ann. Statist. 25 (4) 1661 - 1690, August 1997. https://doi.org/10.1214/aos/1031594736