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Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory

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Author Info

  • Elena Di Bernardino

    ()
    (SAF - Laboratoire de Sciences Actuarielle et Financière - Université Claude Bernard - Lyon I (UCBL) : EA2429)

  • Thomas Laloë

    ()
    (JAD - Laboratoire Jean Alexandre Dieudonné - CNRS : UMR6621 - Université Nice Sophia Antipolis (UNS))

  • Véronique Maume-Deschamps

    ()
    (SAF - Laboratoire de Sciences Actuarielle et Financière - Université Claude Bernard - Lyon I (UCBL) : EA2429)

  • Clémentine Prieur

    ()
    (INRIA Grenoble Rhône-Alpes / LJK Laboratoire Jean Kuntzmann - MOISE - CNRS : UMR5224 - INRIA - Laboratoire Jean Kuntzmann - Université Joseph Fourier - Grenoble I - Institut polytechnique de Grenoble (Grenoble INP))

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    Abstract

    This paper deals with the problem of estimating the level sets of an unknown distribution function $F$. A plug-in approach is followed. That is, given a consistent estimator $F_n$ of $F$, we estimate the level sets of $F$ by the level sets of $F_n$. In our setting no compactness property is a priori required for the level sets to estimate. We state consistency results with respect to the Hausdorff distance and the volume of the symmetric difference. Our results are motivated by applications in multivariate risk theory. In this sense we also present simulated and real examples which illustrate our theoretical results.

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    Bibliographic Info

    Paper provided by HAL in its series Post-Print with number hal-00580624.

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    Date of creation: 08 Feb 2013
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    Publication status: Published, ESAIM: Probability and Statistics, 2013, 17, 236-256
    Handle: RePEc:hal:journl:hal-00580624

    Note: View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/hal-00580624
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    Related research

    Keywords: Level sets ; Distribution function ; Plug-in estimation ; Hausdorff distance ; Conditional Tail Expectation;

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    References

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    1. Embrechts, Paul & Puccetti, Giovanni, 2006. "Bounds for functions of multivariate risks," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 526-547, February.
    2. Dehaan, L. & Huang, X., 1995. "Large Quantile Estimation in a Multivariate Setting," Journal of Multivariate Analysis, Elsevier, vol. 53(2), pages 247-263, May.
    3. Belzunce, F. & Castano, A. & Olvera-Cervantes, A. & Suarez-Llorens, A., 2007. "Quantile curves and dependence structure for bivariate distributions," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 5112-5129, June.
    4. Masse, J. C. & Theodorescu, R., 1994. "Halfplane Trimming for Bivariate Distributions," Journal of Multivariate Analysis, Elsevier, vol. 48(2), pages 188-202, February.
    5. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
    6. Cadre, BenoI^t, 2006. "Kernel estimation of density level sets," Journal of Multivariate Analysis, Elsevier, vol. 97(4), pages 999-1023, April.
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    Cited by:
    1. Cousin, Areski & Di Bernardino, Elena, 2014. "On multivariate extensions of Conditional-Tail-Expectation," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 272-282.

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