Wasserstein Regression
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The analysis of samples of random objects that do not lie in a vector space has found increasing attention in statistics in recent years. An important class of such object data is univariate probability measures defined on the real line. Adopting the Wasserstein metric, we develop a class of regression models for such data, where random distributions serve as predictors and the responses are either also distributions or scalars. To define this regression model, we utilize the geometry of tangent bundles of the metric space of random measures with the Wasserstein metric. The proposed distribution-to-distribution regression model provides an extension of multivariate linear regression for Euclidean data and function-to-function regression for Hilbert space valued data in functional data analysis. In simulations, it performs better than an alternative approach where one first transforms the distributions to functions in a Hilbert space and then applies traditional functional regression. We derive asymptotic rates of convergence for the estimator of the regression coefficient function and for predicted distributions and also study an extension to autoregressive models for distribution-valued time series. The proposed methods are illustrated with data on human mortality and distributions of house prices.
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