We consider the problem of estimating a mean shape from a set of J planar
configurations described by a sequence of k landmarks. We study the consistency
of a smoothed Procrustean mean when the observations obey a deformable model
including some nuisance parameters such as random translations, rotations and
scaling. The main contribution of the paper is to analyze the influence of the
dimension k of the data and of the number J of observed configurations on the
convergence of the smoothed Procrustean estimator to the mean pattern of the
model.

This paper considers the problem of adaptive estimation of a non-homogeneous
intensity function from the observation of n independent Poisson processes
having a common intensity that is randomly shifted for each observed
trajectory. We show that estimating this intensity is a deconvolution problem
for which the density of the random shifts plays the role of the convolution
operator. In an asymptotic setting where the number n of observed trajectories
tends to infinity, we derive upper and lower bounds for the minimax quadratic
risk over Besov balls.

In this paper, we consider the Group Lasso estimator of the covariance matrix
of a stochastic process corrupted by an additive noise. We propose to estimate
the covariance matrix in a high-dimensional setting under the assumption that
the process has a sparse representation in a large dictionary of basis
functions. Using a matrix regression model, we propose a new methodology for
high-dimensional covariance matrix estimation based on empirical contrast
regularization by a group Lasso penalty.

In this paper, we study the problem of nonparametric adaptive estimation of
the covariance function of a stationary Gaussian process. For this purpose, we
consider a wavelet-based method which combines the ideas of wavelet
approximation and estimation by information projection in order to warrants the
positive semidefiniteness property of the solution. The spectral density of the
process is estimated by projecting the wavelet thresholding expansion of the
periodogram onto a family of exponential functions. This ensures that the
spectral density estimator is a strictly positive function.

We propose a model selection approach for covariance estimation of a
multi-dimensional stochastic process. Under very general assumptions, observing
i.i.d replications of the process at fixed observation points, we construct an
estimator of the covariance function by expanding the process onto a collection
of basis functions. We study the non asymptotic property of this estimate and
give a tractable way of selecting the best estimator among a possible set of
candidates. The optimality of the procedure is proved via an oracle inequality
which warrants that the best model is selected.