Techniques are developed for nonparametric analysis of data under a Cox-regression-like model permitting time-dependent covariate effects determined by a regression function $\beta_0(t)$. Estimators resulting from maximization of an appropriate penalized partial likelihood are shown to exist and a computational approach is outlined. Weak uniform consistency (with a rate of convergence) and pointwise asymptotic normality of the estimators are established under regularity conditions. A consistent estimator of a common baseline hazard function is presented and used to construct a consistent estimator of the asymptotic variance of the estimator of the regression function. Extensions to multiple covariates, general relative risk functions and time-dependent covariates are discussed.
"Nonparametric Survival Analysis with Time-Dependent Covariate Effects: A Penalized Partial Likelihood Approach." Ann. Statist. 18 (1) 329 - 353, March, 1990. https://doi.org/10.1214/aos/1176347503