Mixture cure model methodology in survival analysis: Some recent results for the one-sample case

The mixture cure model in survival analysis has received large and growing attention in the last few decades. Restricting ourselves mainly to the one-sample case, we present here an overview drawing together some recent significant advances and earlier results, and pointing out areas where further work is needed. New results presented include a discussion of testing for the presence of long term survivors in the null case (when there are no cures present), the probability that an individual is cured (when cures are present), and further analysis of the idea of sufficient followup. Extreme value methods play a key role. We draw attention to some challenging open problems.