Abstract

The shape parameter, the tail index or the extreme value index (EVI). It plays a very crucial role in the analysis of extremes as it governs the thickness of the distribution right tail. The most popular estimator of the EVI is Hill’s estimator Hill (1975). Csörgo et al. (1985) introduced more general weighs instead of the natural one i/k that appears in the formula of the Hill estimator. The nice properties of the kernel estimator (see, Csörgo et al. (1985)) are smoothness and stability, contrary to Hill’s one which rather exhibits fluctuations along the range of upper extreme values. Both estimators are previous for the complete data. However, in the analysis of lifetime, reliability, or insurance data, the observations are usually randomly censored. In other words, in many real situations, the variable of interest X is not always available.

An appropriate way to model this matter is to introduce a non-negative rv Y; called censoring rv, independent of X and then to consider the rv Z := min (X; Y ) and the indicator variable which determines whether or not X has been observed. Recently, by using a Kaplan-Meier integral Beirlant et al. (2019) proposed a new estimator of the tail index for randomly censored data. Benchaira et al. (2016) proposed a kernel estimator of the tail index for randomly truncated data and established its asymptotic normality.

To the best of our knowledge, when the data are randomly censored, this estimation approach is not yet addressed in the extreme value literature. In this work, we introduce a new kernel estimator to EVI for censored data. This estimator is a generalization of a version of Worms’s estimator (Worms and Worms, 2014) for the indicator function. In other terms, this new kernel estimator is a generalization of the well-known kernel estimator of the extreme value index for complete data introduced by Csörgo et al. (1985), which is a reduced-bias kernel estimator for censored data.