Maximum likelihood estimation for left-censored survival times in an additive hazard model

•Consistency of the maximum likelihood estimator for left-censored survival times.•Asymptotic normality of the maximum likelihood estimator for left-censored survival times.•Statistical inference for left-censored survivals times by concavity of likelihood.•Include time-dependent covariates in additive hazard model.

Motivated by an application from finance, we study randomly left-censored data with time-dependent covariates in a parametric additive hazard model. As the log-likelihood is concave in the parameter, we provide a short and direct proof of the asymptotic normality for the maximal likelihood estimator by applying a result for convex processes from Hjort and Pollard (1993). The technique also yields a new proof for right-censored data. Monte Carlo simulations confirm the nominal level of the asymptotic confidence intervals for finite samples, but also provide evidence for the importance of a proper variance estimator. In the application, we estimate the hazard of credit rating transition, where left-censored observations result from infrequent monitoring of rating histories. Calendar time as time-dependent covariates shows that the hazard varies markedly between years.