Extreme conditional expectile estimation in heavy-tailed heteroscedastic regression models

Stéphane Girard (1) , Gilles Stupfler (2, 3) , Antoine Usseglio-Carleve (1, 2, 4, 3)

1 STATIFY - Modèles statistiques bayésiens et des valeurs extrêmes pour données structurées et de grande dimension
2 ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz]
3 CREST - Centre de Recherche en Economie et Statistique [Bruz]
4 TSE-R - Toulouse School of Economics

Expectiles define a least squares analogue of quantiles. They have been the focus of a substantial quantity of research in the context of actuarial and financial risk assessment over the last decade. The behaviour and estimation of unconditional extreme expectiles using independent and identically distributed heavy-tailed observations has been investigated in a recent series of papers. We build here a general theory for the estimation of extreme conditional expectiles in heteroscedastic regression models with heavy-tailed noise; our approach is supported by general results of independent interest on residual-based extreme value estimators in heavy-tailed regression models, and is intended to cope with covariates having a large but fixed dimension. We demonstrate how our results can be applied to a wide class of important examples, among which linear models, single-index models as well as ARMA and GARCH time series models. Our estimators are showcased on a numerical simulation study and on real sets of actuarial and financial data.

Keywords: Heavy-tailed distribution; Extreme value analysis; Regression models; Expectiles; Residual-based estimators; Single-index model; Heteroscedasticity; Tail empirical process of residuals

Domains: Statistics [math.ST]

Stéphane Girard, Gilles Stupfler, Antoine Usseglio-Carleve. Extreme conditional expectile estimation in heavy-tailed heteroscedastic regression models. Annals of Statistics, In press. ⟨hal-03306230v3⟩