On parameter estimation with the Wasserstein distance

Abstract

Statistical inference can be performed by minimizing, over the parameter space, the Wasserstein distance between model distributions and the empirical distribution of the data. We study asymptotic properties of such minimum Wasserstein distance estimators, complementing results derived by Bassetti, Bodini and Regazzini in 2006. In particular, our results cover the misspecified setting, in which the data-generating process is not assumed to be part of the family of distributions described by the model. Our results are motivated by recent applications of minimum Wasserstein estimators to complex generative models. We discuss some difficulties arising in the approximation of these estimators and illustrate their behavior in several numerical experiments. Two of our examples are taken from the literature on approximate Bayesian computation and have likelihood functions that are not analytically tractable. Two other examples involve misspecified models.

29 pages (+18 pages of appendices), 6 figures. To appear in Information and Inference: A Journal of the IMA. A previous version of this paper contained work on approximate Bayesian computation with the Wasserstein distance, which can now be found at arxiv:1905.03747

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• Minimax Distribution Estimation in Wasserstein Distance
• Optimal transport natural gradient for statistical manifolds with continuous sample space
• The energy distance for ensemble and scenario reduction
• Stability of Gibbs Posteriors from the Wasserstein Loss for Bayesian Full Waveform Inversion