Non asymptotic controls on a recursive superquantile approximation

TL;DR

This paper introduces a recursive stochastic algorithm for joint quantile and superquantile estimation, providing non-asymptotic bounds, $L^p$ controls, and a central limit theorem, advancing statistical estimation methods.

Contribution

It presents a novel recursive algorithm using Cesaro averaging for joint quantile and superquantile estimation with non-asymptotic risk bounds and a CLT.

Findings

Sharp non-asymptotic quadratic risk bounds for superquantile estimator

Non-asymptotic $L^p$ controls for quantile estimation algorithms

Central limit theorem for the joint estimation procedure

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 $L^p$-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.