  We study the problem of estimating a mean pattern from a set of similar
curves in the setting where the variability in the data is due to random
geometric deformations and additive noise. We propose an estimator based on the
notion of Frechet mean that is a generalization of the standard notion of
averaging to non-Euclidean spaces. We derive a minimax rate for this estimation
problem, and we show that our estimator achieves this optimal rate under the
asymptotics where both the number of curves and the number of sampling points
go to infinity.
