Concentration bounds for empirical conditional value-at-risk: The unbounded case

TL;DR

This paper derives new concentration bounds for empirical CVaR estimation from i.i.d. samples of unbounded sub-Gaussian or sub-exponential variables, addressing challenges in risk-sensitive decision making.

Contribution

It introduces a novel one-sided concentration bound for the sample-based CVaR estimator in the unbounded case, leveraging a new concentration result for VaR estimation.

Findings

Provides a concentration bound for CVaR estimator in unbounded settings

Introduces a concentration result for VaR estimator of independent interest

Addresses risk estimation in decision-making under uncertainty

Abstract

In several real-world applications involving decision making under uncertainty, the traditional expected value objective may not be suitable, as it may be necessary to control losses in the case of a rare but extreme event. Conditional Value-at-Risk (CVaR) is a popular risk measure for modeling the aforementioned objective. We consider the problem of estimating CVaR from i.i.d. samples of an unbounded random variable, which is either sub-Gaussian or sub-exponential. We derive a novel one-sided concentration bound for a natural sample-based CVaR estimator in this setting. Our bound relies on a concentration result for a quantile-based estimator for Value-at-Risk (VaR), which may be of independent interest.