Concentration of risk measures: A Wasserstein distance approach

NeurIPS 2019  ·  Prashanth L. A., Sanjay P. Bhat ·

This paper presents a unified approach based on Wasserstein distance to derive concentration bounds for empirical estimates for a broad class of risk measures. The results cover two broad classes of risk measures which are defined in the paper... The classes of risk measures introduced include include as special cases well known risk measures from the finance literature such as conditional value at risk (CVaR), spectral risk measures, utility-based shortfall risk, cumulative prospect theory (CPT) value, and rank dependent expected utility. Two estimation schemes are considered, one for each class of risk measures. One estimation scheme involves applying the risk measure to the empirical distribution function formed from a collection of i.i.d. samples of the random variable (r.v. ), while the second scheme involves applying the same procedure to a truncated sample. The bounds provided apply to three popular classes of distributions, namely sub-Gaussian, sub-exponential and heavy-tailed distributions. The bounds are derived by first relating the estimation error to the Wasserstein distance between the true and empirical distributions, and then using recent concentration bounds for the latter. Previous concentration bounds are available only for specific risk measures such as CVaR and CPT-value. The bounds derived in this paper are shown to either match or improve upon previous bounds in cases where they are available. The usefulness of the bounds is illustrated through an algorithm and the corresponding regret bound for a stochastic bandit problem involving a general risk measure from either of the two classes introduced in the paper. read more

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