Wasserstein regression for distributional data

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 distributions. Adopting the Wasserstein geometry, we develop a class of regression models, for which the predictor and response are both random distributions. The proposed distribution-to-distribution regression model provides an extension of multiple linear regression for Euclidean data and function-to-function regression for Hilbert space valued data in functional data analysis. We derive asymptotic rates of convergence for the estimates of the regression operator and illustrate the proposed method with human mortality data. We also consider an extension to autoregressive modeling of distributional time series and a nonparametric approach when predictors are scalars and responses are distributions.

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