Extreme Value Theory for Cure Models

Survival analysis is a branch of statistics concerned with modeling time-to-event data, where the variable of interest is the time until an event occurs, such as death, disease recurrence, or equipment failure. A key feature of survival data is that it is often right-censored, meaning that for some individuals the event has not occurred by the end of the study. A further complication arises in many applied settings when a subset of the population is effectively immune to the event of interest. For example, in oncology, some patients may be completely cured after treatment and will never experience disease relapse. This leads to so-called cure models, which explicitly account for the presence of a cured fraction in the population. While cure models have received increasing attention in recent years, their methodology remains limited in realistic settings, particularly when the follow-up period of a study is short relative to the time horizon of potential events. Classical cure models often rely on the assumption of sufficient follow-up, meaning that the study duration exceeds the support of the failure time distribution. This is rarely satisfied in practice. The thesis focuses on building the extreme value theory (EVT) framework for cure models in survival analysis and deriving an estimation procedure for the cure rate. The thesis first shows that the tail of the observed sample is driven by the censoring distribution, and therefore does not directly reveal the tail of the uncured population. Then it studies the tail of the uncensored observations and shows that its tail behavior depends on the domain of attraction and endpoint order of the uncured and censoring distributions. Finally, several estimators of the cure rate are proposed and simulations are carried out to analyze their finite sample properties.