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Extremile Regression

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  • Daouia, Abdelaati
  • Gijbels, Irene
  • Stupfler, Gilles

Abstract

Regression extremiles define a least squares analogue of regression quantiles.They are determined by weighted expectations rather than tail probabilities. Of special interest is their intuitive meaning in terms of expected minima and maxima. Their use appears naturally in risk management where, in contrast to quantiles, they fulfill the coherency axiom and take the severity of tail losses into account. In addition, they are comonotonically additive and belong to both the families of spec- tral risk measures and concave distortion risk measures. This paper provides the first detailed study exploring implications of the extremile terminology in a general setting of presence of covariates. We rely on local linear (least squares) check func- tion minimization for estimating conditional extremiles and deriving the asymptotic normality of their estimators. We also extend extremile regression far into the tails of heavy-tailed distributions. Extrapolated estimators are constructed and their asymptotic theory is developed. Some applications to real data are provided.

Suggested Citation

  • Daouia, Abdelaati & Gijbels, Irene & Stupfler, Gilles, 2021. "Extremile Regression," TSE Working Papers 21-1176, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:125140
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    References listed on IDEAS

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    1. Li, Qi & Racine, Jeffrey S, 2008. "Nonparametric Estimation of Conditional CDF and Quantile Functions With Mixed Categorical and Continuous Data," Journal of Business & Economic Statistics, American Statistical Association, vol. 26, pages 423-434.
    2. Abdelaati Daouia & Irène Gijbels & Gilles Stupfler, 2019. "Extremiles: A New Perspective on Asymmetric Least Squares," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(527), pages 1366-1381, July.
    3. Abdelaati Daouia & Laurent Gardes & Stéphane Girard & Alexandre Lekina, 2011. "Kernel estimators of extreme level curves," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(2), pages 311-333, August.
    4. Stéphane Girard & Gilles Stupfler & Antoine Usseglio‐Carleve, 2022. "Nonparametric extreme conditional expectile estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 78-115, March.
    5. Daouia, Abdelaati & Girard, Stéphane & Stupfler, Gilles, 2018. "Tail expectile process and risk assessment," TSE Working Papers 18-944, Toulouse School of Economics (TSE).
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    Cited by:

    1. Chen, Yu & Ma, Mengyuan & Sun, Hongfang, 2023. "Statistical inference for extreme extremile in heavy-tailed heteroscedastic regression model," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 142-162.
    2. Christis Katsouris, 2023. "Quantile Time Series Regression Models Revisited," Papers 2308.06617, arXiv.org, revised Aug 2023.
    3. Christis Katsouris, 2024. "Robust Estimation in Network Vector Autoregression with Nonstationary Regressors," Papers 2401.04050, arXiv.org.
    4. Genest Christian & Scherer Matthias, 2023. "When copulas and smoothing met: An interview with Irène Gijbels," Dependence Modeling, De Gruyter, vol. 11(1), pages 1-16, January.
    5. Marcelo Brutti Righi & Fernanda Maria Muller & Marlon Ruoso Moresco, 2022. "A risk measurement approach from risk-averse stochastic optimization of score functions," Papers 2208.14809, arXiv.org, revised May 2023.

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