Abstract
In this work, we study a new recursive stochastic algorithm for the joint estimation of quantile and superquantile of an unknown distribution. The novelty of this algorithm is to use the Cesaro averaging of the quantile estimation inside the recursive approximation of the superquantile. We provide some sharp non-asymptotic bounds on the quadratic risk of the superquantile estimator for different step size sequences. We also prove new non-asymptotic -controls on the Robbins Monro algorithm for quantile estimation and its averaged version. Finally, we derive a central limit theorem of our joint procedure using the diffusion approximation point of view hidden behind our stochastic algorithm.
Funding Statement
S. G. gratefully acknowledges financial support from the Agence Nationale de la Recherche (MaSDOL grant ANR-19-CE23-0017). S. G. is member of the Institut Universitaire de France (IUF), and part of this work has been carried out with financial support from the IUF. M. C. has been supported by the Chaire “Modélisation Mathématique et Biodiversité” of Veolia Environnement-École Polytechnique-Museum national d’Histoire naturelle-Fondation X. P. C
Acknowledgments
The authors would like to warmly thank Bernard Bercu for fruitful discussions about their work. They also thank the two anonymous referees for their insightful comments and constructive suggestions which helped to improve the paper.
Citation
Manon Costa. Sébastien Gadat. "Non asymptotic controls on a recursive superquantile approximation." Electron. J. Statist. 15 (2) 4718 - 4769, 2021. https://doi.org/10.1214/21-EJS1908
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