  This paper is focused on the statistical analysis of probability measures
$\nu_{1},\ldots,\nu_{n}$ 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 $\nu_{i}$ are absolutely
continuous with densities $f_{i}$ 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 $\nu_{i}$ 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.
