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_14: Estimating an overall density function from repeated observations on each of a sample of independent subjects or experimental units is of interest. An example is provided by biodemographic studies, where one observes age-at-death for several cohorts of flies. Cohorts are kept in separate cages, which form the experimental units. Time variation then is likely to exist between the cohort densities and hazard rates due to cage effects on aging. Given the densities of age-at-death for the individual cohorts, one aims to obtain an estimate for the underlying overall density and hazard rate. In microarray gene expression experiments, similar problems arise when addressing the need for normalization of probe-level data from different arrays. Conventional methods, such as the cross-sectional average density, ignore time variation and hence are often not representative for such data. We view densities as functional data and model individual densities as warped versions of an underlying overall density, where the observed densities are assumed to be realizations of an underlying stochastic process. Quantile-synchronized distribution functions are obtained from an inverse warping mapping, based on quantile synchronization, leading to quantile-synchronized density and hazard functions. Kernel type smoothing methods with plug-in bandwidth selection can be used for estimating the components of the model. Asymptotic properties of the synchronized density estimates are derived. Simulation results show that functional density synchronization is often advantageous when compared to conventional density averaging or simple time-shift warping. Our approach complements previous quantile normalization methods used for microarray expression data and is illustrated with both longevity data obtained for 54 cohorts of mexflies (Mexican fruit flies) and gene expression data of the Ts1Cje mouse study for Down syndrome.
In the one dimensional case ( @math ), computing the Wasserstein barycenter of a finite set of probability measures simply amounts to averaging (in the usual way) their quantile functions. In statistics, this approach has been referred to as quantile synchronization @cite_14 . In the presence of phase variability in the data, quantile synchronization is known to be an appropriate alternative to the usual Euclidean mean of densities to compute a structural mean density that is more consistent with the data. Various asymptotic properties of quantile synchronization are studied in @cite_14 in a statistical model and asymptotic setting similar to that of this paper with @math . However, other measures of risk than the one in this paper are considered in @cite_14 , but the optimality of the resulting convergence rates of quantile synchronization is not discussed.