Approximations to the extreme value index estimator of a heavy-tailed distribution under random censoring and application

We make use of the empirical process theory to approximate the adapted Hill estimator, for censored data, in terms of Gaussian processes. Then, we derive its asymptotic normality, only under the usual second-order condition of regular variation. Our results will be of great interest in establishing the limit distributions of many statistics in extreme value theory under random censoring such as the estimators of tail indices and actuarial risk premiums for heavy-tailed distributions. As an application, we propose an estimator for the excess-of-loss reinsurance premium and establish its asymptotic normality.