  This paper presents a unified approach based on Wasserstein distance to
derive concentration bounds for empirical estimates for two broad classes of
risk measures defined in the paper. The classes of risk measures introduced
include as special cases well known risk measures from the finance literature
such as conditional value at risk (CVaR), optimized certainty equivalent risk,
spectral risk measures, utility-based shortfall risk, cumulative prospect
theory (CPT) value, rank dependent expected utility and distorted risk
measures. 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 each of the two
classes introduced in the paper.
