Estimation of extremes for Weibull-tail distributions in the presence of random censoring

Abstract

The Weibull-tail class of distributions is a sub-class of the Gumbel extreme domain of attraction, and it has caught the attention of a number of researchers in the last decade, particularly concerning the estimation of the so-called Weibull-tail coefficient. In this paper, we propose an estimator of this Weibull-tail coefficient when the Weibull-tail distribution of interest is censored from the right by another Weibull-tail distribution: to the best of our knowledge, this is the first one proposed in this context. A corresponding estimator of extreme quantiles is also proposed. In both mild censoring and heavy censoring (in the tail) settings, asymptotic normality of these estimators is proved, and their finite sample behavior is presented via some simulations.

Nonparametric kernel estimation of Weibull-tail coefficient in presence of the right random censoring

“…Using the abovementioned estimators γY (x) of γ Y (x), let us now consider the estimation of extreme quantiles for Weibull-tail under right random censored data. For any given small probability α n , we can now adapt the classical estimator of q(α n |x) = F ← (α n |x) proposed in Worms & Worms (2019) as follows:…”

“…In some real life applications where the rare events need to be studied, the tail heaviness of the conditional distribution is not verified, particularly in the survival analysis, where the censored data are lifetimes of patients or of animals, or time-to-failure of systems or items. For example, in Gomes & Neves (2011); Worms & Worms (2019), the authors have shown that the datasets of men suffering from a larynx cancer did not exhibit a heavy right-tail.…”

“…This study is motivated by the works of Gardes & Girard (2016), de Wet et al ( 2016) and Worms & Worms (2019). In fact, Gardes & Girard (2016) estimated the conditional tail coefficient of Weibull-type distributions when functional covariate is available.…”

“…de Wet et al (2016) considered the estimation of the tail coefficient of a Weibull-type distribution in the presence of real random covariates. Worms & Worms (2019) proposed an estimator of the Weibull-tail coefficient when the Weibull-tail distribution of interest is censored from the right by another Weibull-tail distribution.…”

Trimmed extreme value estimators for censored heavy-tailed data

“…where is slowly varying at infinity: lim x→∞ (tx) (x) = 1, for every t > 1. Worms and Worms (2019) discuss the analogous problem for Weibull-type distributions. Right-censoring for heavy-tailed distributions in a regression setting was discussed in Ndao et al (2016); Dierckx et al (2019); Goegebeur et al (2019a); Stupfler (2016), while Goegebeur et al (2019b) is on a bivariate extension.…”