We examine a new general class of hazard rate models for survival data, containing a parametric and a nonparametric component. Both can be a mix of a time effect and (possibly time-dependent) marker or covariate effects. A number of well-known models are special cases. In a counting process framework, a general profile likelihood estimator is developed and the parametric component of the model is shown to be asymptotically normal and efficient. The analysis improves on earlier results for special cases. Finite sample properties are investigated in simulations. The estimator is shown to work well under realistic empirical conditions. The estimator is applied to investigate the long-run relationship between birth weight and later-life mortality using data from the Uppsala Birth Cohort Study of individuals born in 1915-1929. The results suggest a relationship that is difficult to capture with simple parametric specifications. Moreover, its shape at higher birth weights differs across gender.