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Title: Statistics of Wasserstein distributionally robust estimators Authors:  Jose Blanchet - Stanford (United States) [presenting]
Abstract: Copulas provide an approach for estimating and characterizing joint distributions. The Wasserstein distance constructs the minimum cost copula (according to a specified criterion) between two marginal distributions. In recent years, a paradigm for robust estimation has emerged based on using this type of copula construction. Given a loss function to be minimized based on an empirical sample, a Wasserstein distributional robust estimator is obtained by choosing a parameter to minimize an expected loss against an adversary (say nature) that wishes to maximize the loss by choosing an appropriate probability model which is coupled with the data according to the Wasserstein distance given a budget constraint. It turns out that by appropriately choosing the loss and the geometry of the Wasserstein distance one can recover many classical statistical estimators (including Lasso, Graphical Lasso, SVM, group Lasso, among many others). A wide range of rich statistical quantities associated with these formulations is studied; for example, the optimal (in a certain sense) choice of the adversarial perturbation, weak convergence of natural confidence regions associated with these formulations, and asymptotic normality of the DRO estimators.