An essential aspect of survival analysis is the estimation and prediction of survival probabilities for individuals. For this purpose, mathematical modelling of the hazard rate function is a fundamental issue. This thesis focuses on the novel estimation and application of hazard rate functions in mathematical and medical research. In mathematical research we focus on the development of a semiparametric kernel-based estimate of hazard rate function and a L1 error optimal kernel hazard rate estimate. In medical research we concentrate on the development and validation of survival models using individual participant data from multiple studies. We also consider how to fit survival models that predict individual response to treatment effectiveness, given IPD from multiple trials
