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Improved Confidence Intervals for Expectiles

Author

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  • Spiridon Penev

    (Department of Statistics, The University of New South Wales Sydney, Kensington, NSW 2052, Australia
    These authors contributed equally to this work.)

  • Yoshihiko Maesono

    (Department of Mathematics, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
    These authors contributed equally to this work.)

Abstract

Expectiles were introduced to statistics around 40 years ago, but have recently gained renewed interest due to their relevance in financial risk management. In particular, the 2007–2009 global financial crisis highlighted the need for more robust risk evaluation tools, leading to the adoption of inference methods for expectiles. While first-order asymptotic inference results for expectiles are well established, higher-order asymptotic results remain underdeveloped. This study aims to fill that gap by deriving higher-order asymptotic results for expectiles, ultimately improving the accuracy of confidence intervals. The paper outlines the derivation of the Edgeworth expansion for both the standardized and studentized versions of the kernel-based estimator of the expectile, using large deviation results on U -statistics. The expansion is then inverted to construct more precise confidence intervals for the expectile. These theoretical results were applied to moderate sample sizes ranging from 20 to 200. To demonstrate the advantages of this methodology, an example from risk management is presented. The enhanced confidence intervals consistently outperformed those based on the first-order normal approximation. The methodology introduced in this paper can also be extended to other contexts.

Suggested Citation

  • Spiridon Penev & Yoshihiko Maesono, 2025. "Improved Confidence Intervals for Expectiles," Mathematics, MDPI, vol. 13(3), pages 1-27, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:3:p:510-:d:1583031
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    References listed on IDEAS

    as
    1. Gneiting, Tilmann, 2011. "Making and Evaluating Point Forecasts," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 746-762.
    2. Yoshihiko Maesono & Spiridon Penev, 2013. "Improved confidence intervals for quantiles," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(1), pages 167-189, February.
    3. 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.
    4. Bellini, Fabio, 2012. "Isotonicity properties of generalized quantiles," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 2017-2024.
    5. Yoshihiko Maesono & Spiridon Penev, 2011. "Edgeworth expansion for the kernel quantile estimator," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(3), pages 617-644, June.
    6. 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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