Semiparametric inference for complex and structural models in survival analysis

Survival analysis examines and models the time it takes for events to occur. The typical event is death, from which the name `survival analysis' and much of its terminology derives. Since the data can only be collected over a finite period of time, the `time to event' may not be observed for all the individuals. This is the case for example when a patient leaves a clinical study before it ends or she/he is still alive by the end of the study. In such a case, the death time (time to event) for this individual is unknown. Such a phenomenon, named censoring, creates some unusual difficulties in the analysis of survival data that cannot be handled properly by standard statistical methods. In traditional survival analysis, all subjects in the population are assumed to be susceptible to the event of interest, that is, every subject has either already experienced the event or will experience it in the future. However in many situations it may happen that a fraction of individuals (long-term survivors) will never experience the event, that is, they are considered to be event free. For example, a treatment is assigned to patients in order to evaluate the effect on the recurrence of a disease. Many individuals never experience recurrences and thus can be regarded as cured or immune individuals. We refer to the book of Maller and Zhou (1996) for an overview on the topic of cure models in survival analysis. An important drawback related to many existing cure models in the literature, is that they assume that the follow-up period of the study is long enough so that all censored observations in the right tail of the distribution can be considered as cured. This is however a heavy assumption in practice, as often the study has to be stopped earlier for financial and/or practical reasons. In this thesis we wish to investigate ways to overcome this model assumption. We will do that in a framework where apart from the survival time, we also observe a number of covariates for each subject. A possible way out is to make use of so called location-scale regression models for the uncured individuals. In these models it is possible to identify the tail of the conditional distribution as soon as there is a region of the covariate space in which the follow-up is sufficiently long. We will study the identifiability of this model in a formal way. Once the identifiability is established, we will propose an estimation method for this model, which will be based on flexible parametric families, and we will carry out finite sample simulations to compare the performance of the new procedure with existing competitors in the literature.