Let P 2,ac be the set of Borel probabilities on R d with finite second moment and absolutely continuous with respect to Lebesgue measure. We consider the problem of finding the barycenter (or Fréchet mean) of a finite set of probabilities ν 1 , . . . , ν k ∈ P 2,ac with respect to the L 2 −Wasserstein metric. For this task we introduce an operator on P 2,ac related to the optimal transport maps pushing forward any µ ∈ P 2,ac to ν 1 , . . . , ν k . Under very general conditions we prove that the barycenter must be a fixed point for this operator and introduce an iterative procedure which consistently approximates the barycenter. The procedure allows effective computation of barycenters in any location-scatter family, including the Gaussian case. In such cases the barycenter must belong to the family, thus it is characterized by its mean and covariance matrix. While its mean is just the weighted mean of the means of the probabilities, the covariance matrix is characterized in terms of their covariance matrices Σ 1 , . . . , Σ k through a nonlinear matrix equation. The performance of the iterative procedure in this case is illustrated through numerical simulations, which show fast convergence towards the barycenter.
1 Although i.i.d.-ness can be relaxed into exchangeability, we are sticking to the former. 2 Typical examples are linear models, with Z i (θ θ θ) = X i − c i θ θ θ (c i a q-vector of covariates and θ θ θ ∈ R q ), or firstorder autoregressive models, with Z i (θ ) = X i − θX i−1 (where i denotes time and θ ∈ (−1, 1); see, for example, Hallin and Werker (1999)), etc.3 Those ranks indeed are maximal invariant under the group of continuous monotone increasing transformations of Z 1 (θ θ θ 0 ), . . . , Z n (θ θ θ 0 ); see, for instance, Example 7 in Lehmann and Scholz (1992).
Weighted L 2 functionals of the empirical quantile process appear as a component of many test statistics, in particular in tests of fit to location-scale families of distributions based on weighted Wasserstein distances. An essentially complete set of distributional limit theorems for the squared empirical quantile process integrated with respect to general weights is presented. The results rely on limit theorems for quadratic forms in exponential random variables, and the proofs use only simple asymptotic theory for probability distributions in R n . The limit theorems are then applied to determine the asymptotic distribution of the test statistics on which weighted Wasserstein tests are based. In particular, this paper contains an elementary derivation of the limit distribution of the Shapiro-Wilk test statistic under normality.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.