Extreme Kaplan-Meier integrals

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 among various tail regimes. A noteworthy by-product is the extension of generalized Hill-type estimators of extremes to encompass arbitrary tail behavior, which is of independent interest. The theoretical framework is applied to construct novel estimators for real-valued extreme value indices for right-censored data. Simulation studies confirm the asymptotic results and, in a competitor case, mostly show superiority in mean square error. An application to brain cancer data demonstrates that censoring effects are properly accounted for, even when focusing solely on tail classification.