Abstract
This paper is devoted to two different two-time-scale stochastic approximation algorithms for superquantile, also known as conditional value-at-risk, estimation. We shall investigate the asymptotic behavior of a Robbins-Monro estimator and its convexified 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.
Highlights
Estimating quantiles has a longstanding history in statistics and probability
We focus our attention on the asymptotic normality of our two-time-scale stochastic algorithms (2.1) and (2.3)
We focus our attention on the proof of the law of iterated logarithm (LIL) given by (5.57)
Summary
Estimating quantiles has a longstanding history in statistics and probability. Except in parametric models where explicit formula are available, the estimation of quantiles is a real issue. The most common way to estimate quantiles is to make use of order statistics, see among other references [1, 13]. Another strategy is to make use of stochastic approximation algorithms and the pioneering work in this vein is the celebrated paper by Robbins and Monro [23]. One can observe that the superquantile provides more information on the tail of the distribution of the random variable X. We propose the almost sure convergence of two-time-scale stochastic approximation algorithms for superquantile estimation.
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