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Stochastic approximation algorithms for superquantiles estimation

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  • Costa, Manon
  • Gadat, Sébastien
  • Bercu, Bernard

Abstract

This paper is devoted to two dierent two-time-scale stochastic ap- proximation algorithms for superquantile estimation. We shall investigate the asymptotic behavior of a Robbins-Monro estimator and its convexied version. Our main contribution is to establish the almost sure convergence, the quadratic strong law and the law of iterated logarithm for our estimates via a martingale approach. A joint asymptotic normality is also provided. Our theoretical analysis is illustrated by numerical experiments on real datasets.

Suggested Citation

  • Costa, Manon & Gadat, Sébastien & Bercu, Bernard, 2020. "Stochastic approximation algorithms for superquantiles estimation," TSE Working Papers 20-1142, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:124668
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    References listed on IDEAS

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    1. Gadat, Sébastien & Panloup, Fabien & Saadane, Sofiane, 2016. "Stochastic Heavy Ball," TSE Working Papers 16-712, Toulouse School of Economics (TSE).
    2. Bardou O. & Frikha N. & Pagès G., 2009. "Computing VaR and CVaR using stochastic approximation and adaptive unconstrained importance sampling," Monte Carlo Methods and Applications, De Gruyter, vol. 15(3), pages 173-210, January.
    3. Pelletier, Mariane, 1998. "On the almost sure asymptotic behaviour of stochastic algorithms," Stochastic Processes and their Applications, Elsevier, vol. 78(2), pages 217-244, November.
    4. Aharon Ben-Tal & Marc Teboulle, 1986. "Expected Utility, Penalty Functions, and Duality in Stochastic Nonlinear Programming," Management Science, INFORMS, vol. 32(11), pages 1445-1466, November.
    5. Bercu, B., 2004. "On the convergence of moments in the almost sure central limit theorem for martingales with statistical applications," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 157-173, May.
    6. Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional value-at-risk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1443-1471, July.
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    Citations

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    Cited by:

    1. Gadat, Sébastien & Gavra, Ioana, 2021. "Asymptotic study of stochastic adaptive algorithm in non-convex landscape," TSE Working Papers 21-1175, Toulouse School of Economics (TSE).
    2. Vishwajit Hegde & Arvind S. Menon & L. A. Prashanth & Krishna Jagannathan, 2021. "Online Estimation and Optimization of Utility-Based Shortfall Risk," Papers 2111.08805, arXiv.org, revised Nov 2023.
    3. Manon Costa & Sébastien Gadat, 2021. "Non-asymptotic study of a recursive superquantile estimation algorithm," Post-Print hal-03610477, HAL.
    4. Sébastien Gadat & Ioana Gavra, 2022. "Asymptotic study of stochastic adaptive algorithm in non-convex landscape," Post-Print hal-03857182, HAL.
    5. Gadat, Sébastien & Costa, Manon, 2020. "Non asymptotic controls on a stochastic algorithm for superquantile approximation," TSE Working Papers 20-1149, Toulouse School of Economics (TSE).

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