Electronic Journal of Statistics

Upper and lower risk bounds for estimating the Wasserstein barycenter of random measures on the real line

Jérémie Bigot, Raúl Gouet, Thierry Klein, and Alfredo López

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Abstract

This paper is focused on the statistical analysis of probability measures $\boldsymbol{\nu }_{1},\ldots ,\boldsymbol{\nu }_{n}$ on ${\mathbb{R}}$ that can be viewed as independent realizations of an underlying stochastic process. We consider the situation of practical importance where the random measures $\boldsymbol{\nu }_{i}$ are absolutely continuous with densities $\boldsymbol{f}_{i}$ that are not directly observable. In this case, instead of the densities, we have access to datasets of real random variables $(X_{i,j})_{1\leq i\leq n;\;1\leq j\leq p_{i}}$ organized in the form of $n$ experimental units, such that $X_{i,1},\ldots ,X_{i,p_{i}}$ are iid observations sampled from a random measure $\boldsymbol{\nu }_{i}$ for each $1\leq i\leq n$. In this setting, we focus on first-order statistics methods for estimating, from such data, a meaningful structural mean measure. For the purpose of taking into account phase and amplitude variations in the observations, we argue that the notion of Wasserstein barycenter is a relevant tool. The main contribution of this paper is to characterize the rate of convergence of a (possibly smoothed) empirical Wasserstein barycenter towards its population counterpart in the asymptotic setting where both $n$ and $\min_{1\leq i\leq n}p_{i}$ may go to infinity. The optimality of this procedure is discussed from the minimax point of view with respect to the Wasserstein metric. We also highlight the connection between our approach and the curve registration problem in statistics. Some numerical experiments are used to illustrate the results of the paper on the convergence rate of empirical Wasserstein barycenters.

Article information

Source
Electron. J. Statist., Volume 12, Number 2 (2018), 2253-2289.

Dates
Received: March 2017
First available in Project Euclid: 23 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1532333005

Digital Object Identifier
doi:10.1214/18-EJS1400

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties

Keywords
Wasserstein space Fréchet mean barycenter of probability measures functional data analysis density registration phase and amplitude variability smoothing minimax optimality

Rights
Creative Commons Attribution 4.0 International License.

Citation

Bigot, Jérémie; Gouet, Raúl; Klein, Thierry; López, Alfredo. Upper and lower risk bounds for estimating the Wasserstein barycenter of random measures on the real line. Electron. J. Statist. 12 (2018), no. 2, 2253--2289. doi:10.1214/18-EJS1400. https://projecteuclid.org/euclid.ejs/1532333005


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