A1322
Title: On the attainment of the Wasserstein-Cramer-Rao lower bound and the location scale family
Authors: Takeru Matsuda - University of Tokyo (Japan)
Hayato Nishimori - The University of Tokyo (Japan) [presenting]
Abstract: In information geometry, the Fisher information is regarded as a Riemannian metric that defines the local distance structure in the space of probability distributions. It also gives a lower bound on the variance of (unbiased) estimators in the Cramer-Rao inequality. On the other hand, it is recently reported that the Wasserstein distance, which is the optimal transportation cost between distributions, induces another Riemannian metric and an analogous inequality called the Wasserstein-Cramer-Rao inequality holds. The Wasserstein metric is obtained explicitly in statistical models on the real line by parametrizing the bijection that gives the push-forward measure instead of parametrizing the probability density directly. This parametrized bijection also gives the metric in multivariate models whose copulas do not depend on the parameter. In addition, the first and second-order moment estimators attain the Wasserstein-Cramer-Rao lower bound if and only if the statistical model is the location-scale family. Furthermore, if the statistical model is the location-scale family, estimators of the mean and variance asymptotically attain the lower bound.
