This paper is focused on the statistical analysis of probability measures @math on @math that can be viewed as independent realizations of an underlying stochastic process. We consider the situation of practical importance where the random measures @math are absolutely continuous with densities @math that are not directly observable. In this case, instead of the densities, we have access to datasets of real random variables @math organized in the form of @math experimental units, such that @math are iid observations sampled from a random measure @math for each @math . 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 @math and @math 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.
@cite_2: We develop a canonical framework for the study of the problem of registration of multiple point processes subjected to warping, known as the problem of separation of amplitude and phase variation. The amplitude variation of a real random function Y(x) : x is an element of [0, 1] corresponds to its random oscillations in the y-axis, typically encapsulated by its (co) variation around a mean level. In contrast, its phase variation refers to fluctuations in the x-axis, often caused by random time changes. We formalise similar notions for a point process, and nonparametrically separate them based on realisations of i.i.d. copies Pi(i) of the phase-varying point process. A key element in our approach is to demonstrate that when the classical phase variation assumptions of Functional Data Analysis (FDA) are applied to the point process case, they become equivalent to conditions interpretable through the prism of the theory of optimal transportation of measure. We demonstrate that these induce a natural Wasserstein geometry tailored to the warping problem, including a formal notion of bias expressing over-registration. Within this framework, we construct nonparametric estimators that tend to avoid over-registration in finite samples. We show that they consistently estimate the warp maps, consistently estimate the structural mean, and consistently register the warped point processes, even in a sparse sampling regime. We also establish convergence rates, and derivev root n-consistency and a central limit theorem in the Cox process case under dense sampling, showing rate optimality of our structural mean estimator in that case.
The results of this paper are very much connected with those in @cite_2 where a new framework is developed for the registration of multiple point processes on the real line for the purpose of separating amplitude and phase variation in such data. In @cite_2 , consistent estimators of the structural mean of multiple point processes are obtained by the use of smoothed Wasserstein barycenters with an appropriate choice of kernel smoothing. Also, rates of convergence of such estimators are derived for the Wasserstein metric. The statistical analysis of multiple point processes is very much connected to the study of repeated observations organized in samples from independent subjects or experimental units. Therefore, some of our results in this paper on smoothed empirical Wasserstein barycenters are built upon the work in @cite_2 . Nevertheless, novel contributions include the derivation of an exact formula to compute the risk of non-smoothed Wasserstein barycenters in the case of samples of equal size, and new upper bounds on the rate of convergence of the Wasserstein risk of non-smoothed and smoothed empirical Wasserstein barycenters, together with a discussion of their optimality from the minimax point of view.