This paper presents new approaches for the estimation of the extreme value index in the framework of randomly censored (from the right) samples, based on the ideas of Kaplan-Meier integration and the synthetic data approach of S.Leurgans (1987). These ideas are developed here in the heavy tail case and for the adaptation of the Hill estimator, for which the consistency is proved under first order conditions. Simulations show good performance of the two approaches, with respect to the only existing adaptation of the Hill estimator in this context.

Let F be a distribution function in the domain of attraction of an extreme-value distribution H γ. If F u is the distribution function of the excesses over u and G γ the distribution function of the generalized Pareto distribution, then it is well known that F u (x) converges to G γ(x/σ(u)) as u tends to the end point of F, where σ is an appropriate normalizing function. We study the rate of (uniform) convergence to 0 of F̅ u (x)-G̅γ((x+u-α(u))/σ(u)), where α and σ are two appropriate normalizing functions.

We revisit the estimation of the extreme value index for randomly censored data from a heavy tailed distribution. We introduce a new class of estimators which encompasses earlier proposals given in Worms and Worms (2014) and Beirlant et al. (2018), which were shown to have good bias properties compared with the pseudo maximum likelihood estimator proposed in Beirlant et al. (2007) and Einmahl et al. (2008). However the asymptotic normality of the type of estimators first proposed in Worms and Worms (2014) was still lacking. We derive an asymptotic representation and the asymptotic normality of the larger class of estimators and consider their finite sample behaviour. Special attention is paid to the case of heavy censoring, i.e. where the amount of censoring in the tail is at least 50%. We obtain the asymptotic normality with a classical ? k rate where k denotes the number of top data used in the estimation, depending on the degree of censoring.