Nonparametric estimation of the conditional extreme-value index with random covariates and censoring

Abstract

International audienceEstimation of the extreme-value index of a heavy-tailed distribution is addressed when some random covariate information is available and the data are randomly right-censored. An inverse-probability-of-censoring-weighted kernel version of Hill's estimator of the extreme-value index is proposed and its asymptotic normality is established. Based on this, a Weissman-type estimator of conditional extreme quantiles is also constructed. A simulation study is conducted to assess the finite-sample behaviour of the proposed estimators

On the study of extremes with dependent random right-censoring

“…A similar idea, albeit implemented differently, is used by Beirlant et al (2007). This line of thought was then followed by Gomes and Neves (2011), Ndao et al (2014Ndao et al ( , 2016, Brahimi et al (2015), Beirlant et al (2016) and Stupfler (2016) in their respective contexts. Our aim in Section 3 below is to carry out an analogue study in our dependent censoring context and see what influence the introduction of dependence in the censoring mechanism, via an extreme value copula, has on the distribution of (Z, δ) given that Z is large.…”

“…For ease of exposition, we work here under the conditions of Theorem 2 and we further assume that γ Y γ T > 0. The estimators of Beirlant et al (2007Beirlant et al ( , 2016, Einmahl et al (2008), Gomes and Neves (2011), Ndao et al (2014Ndao et al ( , 2016, Brahimi et al (2015) and Stupfler (2016) are all based on the fact that in the independent censoring case,…”

“…In the aforementioned examples, this would ultimately lead to the analysis of survival times of exceptionally strong/weak patients to a given disease, extreme losses/payouts in insurance, or failure times for highly resistant/unreliable devices. This subfield of extreme value statistics has received a good amount of attention in recent years: we refer to Beirlant et al (2007Beirlant et al ( , 2010Beirlant et al ( , 2016, Einmahl et al (2008), Gomes and Neves (2011), Ndao et al (2014Ndao et al ( , 2016, Sayah et al (2014), Worms and Worms (2014), Brahimi et al (2015) and Stupfler (2016). All of these papers work under the hypothesis that the variable of interest Y and the censoring variable T are independent random variables; in the case of Ndao et al (2014Ndao et al ( , 2016 and Stupfler (2016), the condition is actually conditional independence given a suitable, in practice low-dimensional, covariate.…”

“…The integration of valuable, preferably high-dimensional covariate information may also be helpful if conditional independence given the covariate is reasonable (see Zeng 2004, Li et al 2007and Hsu and Taylor 2010. Let us point out that the studies by Ndao et al (2014Ndao et al ( , 2016 and Stupfler (2016) did consider incorporating a low-dimensional covariate X, but the common idea underpinning these papers is to estimate the conditional extreme value index of Y given X = x by adapting the procedure of Einmahl et al (2008), developed for independent censoring, using kernel-type techniques. The introduction of covariate information is therefore not motivated by a reduction of the dependence between Y and T ; in fact, these papers ignore altogether the issue of dependence and its consequences upon the inference about the extremes of the variable of interest.…”

On a relationship between randomly and non-randomly thresholded empirical average excesses for heavy tails

“…Several studies have since then proposed alternative techniques for tail index estimation in this context; we refer to Benchaira et al (2016a, b), Worms and Worms (2016) and Haouas et al (2017). The random right-truncation context should not be mistaken for random right-censoring, where the available information is made of the pairs (min(Y i , T i ), 1 {Y i ≤T i } ), 1 ≤ i ≤ n. The latter context has received a substantial amount of attention over the last decade: we refer to Beirlant et al (2007Beirlant et al ( , 2010Beirlant et al ( , 2016, Einmahl et al (2008), Gomes andNeves (2011), Ndao et al (2014), Sayah et al (2014), Worms and Worms (2014), Brahimi et al (2015), Ndao et al (2016), Stupfler (2016), Dierckx et al (2018) and Stupfler (2019).…”