  Wasserstein distributionally robust optimization estimators are obtained as
solutions of min-max problems in which the statistician selects a parameter
minimizing the worst-case loss among all probability models within a certain
distance (in a Wasserstein sense) from the underlying empirical measure. While
motivated by the need to identify optimal model parameters or decision choices
that are robust to model misspecification, these distributionally robust
estimators recover a wide range of regularized estimators, including
square-root lasso and support vector machines, among others, as particular
cases. This paper studies the asymptotic normality of these distributionally
robust estimators as well as the properties of an optimal (in a suitable sense)
confidence region induced by the Wasserstein distributionally robust
optimization formulation. In addition, key properties of min-max
distributionally robust optimization problems are also studied, for example, we
show that distributionally robust estimators regularize the loss based on its
derivative and we also derive general sufficient conditions which show the
equivalence between the min-max distributionally robust optimization problem
and the corresponding max-min formulation.
