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Extremile Regression

Author

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  • Abdelaati Daouia
  • Irène Gijbels
  • Gilles Stupfler

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 spectral risk measures and concave distortion risk measures. This article 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 function 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

  • Abdelaati Daouia & Irène Gijbels & Gilles Stupfler, 2022. "Extremile Regression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 117(539), pages 1579-1586, September.
    • Handle: RePEc:taf:jnlasa:v:117:y:2022:i:539:p:1579-1586
    DOI: 10.1080/01621459.2021.1875837
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    1. Gilbert W. Bassett, 2004. "Pessimistic Portfolio Allocation and Choquet Expected Utility," Journal of Financial Econometrics, Oxford University Press, vol. 2(4), pages 477-492.
    2. Abdelaati Daouia & Irène Gijbels & Gilles Stupfler, 2022. "Extremile Regression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 117(539), pages 1579-1586, September.
    3. Bruce Jones & Ričardas Zitikis, 2003. "Empirical Estimation of Risk Measures and Related Quantities," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 44-54.
    4. Wang, Shaun, 1996. "Premium Calculation by Transforming the Layer Premium Density," ASTIN Bulletin, Cambridge University Press, vol. 26(1), pages 71-92, May.
    5. 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.
    6. Hayfield, Tristen & Racine, Jeffrey S., 2008. "Nonparametric Econometrics: The np Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 27(i05).
    7. Gilbert W. Bassett Jr Bassett & Roger Koenker & Gregory Kordas, 2004. "Pessimistic portfolio allocation and Choquet expected utility," CeMMAP working papers 09/04, Institute for Fiscal Studies.
    8. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    9. Acerbi, Carlo, 2002. "Spectral measures of risk: A coherent representation of subjective risk aversion," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1505-1518, July.
    10. 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.
    11. 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.
    12. 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.
    13. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    14. Robert Serfling, 2002. "Quantile functions for multivariate analysis: approaches and applications," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 56(2), pages 214-232, May.
    15. Daouia, Abdelaati & Girard, Stéphane & Stupfler, Gilles, 2018. "Tail expectile process and risk assessment," TSE Working Papers 18-944, Toulouse School of Economics (TSE).
    16. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-847, July.
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    2. 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.
    3. Abdelaati Daouia & Irène Gijbels & Gilles Stupfler, 2022. "Extremile Regression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 117(539), pages 1579-1586, September.
    4. Christis Katsouris, 2023. "Quantile Time Series Regression Models Revisited," Papers 2308.06617, arXiv.org, revised Aug 2023.
    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.
    6. Christis Katsouris, 2024. "Robust Estimation in Network Vector Autoregression with Nonstationary Regressors," Papers 2401.04050, arXiv.org.

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