An inverse problem point of view for adaptive estimation in a shifted curves model

**Jérémie Bigot**

**Université de Toulouse, France**

In this talk, we
focus on the problem of adaptive estimation of a mean pattern in a
randomly
shifted curve model. We show that this problem can be transformed into
a linear inverse
problem, where the density of the random shifts plays the role of a
convolution operator.
An adaptive estimator of the common shape, based on wavelet
thresholding is proposed. We study its consistency for the quadratic
risk as the number of observed curves tends to
infinity, and this estimator is shown to achieve near-minimax rate of
convergence over a
large class of Besov balls. This rate depends both on the smoothness of
the common shape
of the curves and on the decay of the Fourier coefficients of the
density of the random
shifts. Hence, this paper makes a connection between mean pattern
estimation and the
statistical analysis of linear inverse problems, which is a new point
of view on curve
registration and image warping problems. Some numerical experiments are
given to
illustrate the performances of our approach and to compare them with
another algorithm
existing in the literature.