This talk is intended to illustrate some of the very interesting problems that arise in the study of survival analysis data from an applied, a mathematical and a statistical ("perspective") perspecitve. From a practical point of view, a difficulty in the survival analysis of medical research data is that the observation of a failure time (such as recurrence of disease or death) may be made impossible (too many "by") by by the occurrence of a censoring event such as termination of the study or withdrawal from the study. Such data is said to be right censored and we assume the censoring times to be random.
Gill(1980) in his work on censoring and stochastic integrals showed that the mathematically rigorous way to view these problems is by using counting processes and martingale theory. Here we will concentrate on nonparametric estimation of the hazard function assuming some smoothness properties. From a decision theoretic perspective in ("statistics") statistcs we are often interested in optimality properties of estimators such as minimax. Kernel smoothing as well as wavelet estimation will be discussed in this context. It will also be shown that wavelet modeling of the hazard function can sometimes lead to a simple parametric model. The methods are illustrated on some interesting data sets.