Nonparametric estimation of the conditional tail index and extreme quantiles under random censoring

Abstract

23International audienceIn this paper, we investigate the estimation of the tail index and extreme quantiles of a heavy-tailed distribution when some covariate information is available and the data are randomly right-censored. We construct several estimators by combining a moving-window technique (for tackling the covariate information) and the inverse probability-of-censoring weighting method, and we establish their asymptotic normality. A comprehensive simulation study is conducted to evaluate the finite-sample performance of the proposed estimators and to identify their application scope

A Simulation Comparison of Estimators of Conditional Extreme Value Index under Right Random Censoring

“…Overall, we found that the performance of estimators depend on the distribution, the number of top order statistics and the percentage of censoring. Contrary to what was reported in Ndao et al (2014), we found that the Hill estimator has large bias and MSE except for samples generated from the Fréchet distribution. The proposed PPD estimator is universally competitive in estimating γ 1 (x) regardless of its size, percentage of censoring and number of top order statistics.…”

“…The existing estimators result from the application of the moving window technique (Gardes and Girard, 2008) and the inverse probability-of-censoring weighted method (Beirlant et al, 2007;Einmahl et al, 2008) to adapt classical estimators to censoring. Ndao et al (2014) used this approach to adapt the Hill, generalised Hill and moment estimators to censoring. In this section, we review these estimators and follow a similar approach to propose other estimators of the extreme value index in the next section.…”

“…In this section, we propose some estimators for the estimation of conditional extreme value index when observations are subject to right random censoring. We follow closely the methodology in Ndao et al (2014). Firstly, the estimators are presented to take into account the covariate and are subsequently adapted to censoring.…”

“…In particular, the Hill, generalised Hill and moment type estimators were proposed using the moving window method (Gardes and Girard, 2008) and adapting the estimators to censoring by utilising the inverse probability-of-censoring weighted method (Beirlant et al, 2007;Einmahl et al, 2008). Unlike Ndao et al (2014), Stupfler (2016) proposed a moment estimator valid for all domains of attraction. In addition, Ndao et al (2016) addressed the estimation of the extreme value index under censoring and the presence of random covariates.…”

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

“…Matthys et al (2004), Beirlant et al (2007) and Einmahl et al (2008) additionally address estimation of extreme quantiles. Ndao et al (2014) address estimation of the conditional extreme-value index and conditional extreme quantiles with fixed covariates and censoring. To our knowledge, estimation of the conditional extreme-value index and extreme quantiles with random covariates and censoring has not yet been addressed.…”