Estimation of the extreme value index in a censorship framework: Asymptotic and finite sample behavior

Abstract

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.

Trimmed extreme value estimators for censored heavy-tailed data

“…In Einmahl et al (2008) the asymptotics for H k = H K0 k was discussed in detail. Beirlant et al (2019) provided an asymptotic normality result for H W k when p > 1/2, but that estimator is not in the current kernel framework. Here we provide asymptotic representations for the class of kernel estimators in the form H K k .…”

“…Here we provide asymptotic representations for the class of kernel estimators in the form H K k . To this end, we make use of second-order assumptions which were first proposed in Hall and Welsh (1985) and which have widely been used in papers on the estimation of the extreme value index for Pareto-type distributions both in the non-censoring case such as Csörgő et al (1985) and the censoring case as in Beirlant et al (2019):…”

“…Most efforts however have been dedicated to heavy-tailed distributions, with several papers being motivated by heavy-tailed insurance claim data with long development times of the claims, see e.g. Beirlant et al (2010Beirlant et al ( , 2016Beirlant et al ( , 2018Beirlant et al ( , 2019; Worms and Worms (2014, 2016; Ndao et al (2014). The underlying model assumption here is that the random variable of interest X has a Pareto-type distribution function…”

“…Beirlant et al (2018), the estimator H W k is known to exhibit superior behaviour in comparison with H k , especially with respect to bias. The mathematical treatment of this estimator turns out to be difficult and the asymptotic normality of H W k has only be derived in Beirlant et al (2019) under light censoring, i.e. ξ < ξ c or p > 1/2, and some regularity conditions.…”

Functional kernel estimation of the conditional extreme value index under random right censoring

“…While Worms and Worms (2018) proposed estimator of the extreme value index by considering heavy tailed lifetime data under random censoring and competing risks, using the Aalen-Johansen estimator of the cumulative incidence function. Beirlant et al(2019) introduced a new class of estimator which generalized the proposed estimator of Worms and Worms (2014) and Beirlant et al(2018) Some important literature is devoted to the estimation of the conditional quantile of a scalar response given a functional covariate. Gardes and Girard (2012) dealt with the estimation of conditional quantiles when the covariate is functional and when the order of the quantiles converges to one as the sample size increases.…”