  A novel and comprehensive methodology designed to tackle the challenges posed
by extreme values in the context of random censorship is introduced. The main
focus is on the analysis of integrals based on the product-limit estimator of
normalized upper order statistics, called extreme Kaplan--Meier integrals.
These integrals allow for the transparent derivation of various important
asymptotic distributional properties, offering an alternative approach to
conventional plug-in estimation methods. Notably, this methodology demonstrates
robustness and wide applicability within the scope of max-domains of
attraction. A noteworthy by-product is the extension of generalized Hill-type
estimators of extremes to encompass all max-domains of attraction, which is of
independent interest. The theoretical framework is applied to construct novel
estimators for positive and real-valued extreme value indices for
right-censored data. Simulation studies supporting the theory are provided.
