62 research outputs found
Adaptive Covariance Estimation with model selection
We provide in this paper a fully adaptive penalized procedure to select a
covariance among a collection of models observing i.i.d replications of the
process at fixed observation points. For this we generalize previous results of
Bigot and al. and propose to use a data driven penalty to obtain an oracle
inequality for the estimator. We prove that this method is an extension to the
matricial regression model of the work by Baraud
General model selection estimation of a periodic regression with a Gaussian noise
This paper considers the problem of estimating a periodic function in a
continuous time regression model with an additive stationary gaussian noise
having unknown correlation function. A general model selection procedure on the
basis of arbitrary projective estimates, which does not need the knowledge of
the noise correlation function, is proposed. A non-asymptotic upper bound for
quadratic risk (oracle inequality) has been derived under mild conditions on
the noise. For the Ornstein-Uhlenbeck noise the risk upper bound is shown to be
uniform in the nuisance parameter. In the case of gaussian white noise the
constructed procedure has some advantages as compared with the procedure based
on the least squares estimates (LSE). The asymptotic minimaxity of the
estimates has been proved. The proposed model selection scheme is extended also
to the estimation problem based on the discrete data applicably to the
situation when high frequency sampling can not be provided
Adaptive density estimation for stationary processes
We propose an algorithm to estimate the common density of a stationary
process . We suppose that the process is either or
-mixing. We provide a model selection procedure based on a generalization
of Mallows' and we prove oracle inequalities for the selected estimator
under a few prior assumptions on the collection of models and on the mixing
coefficients. We prove that our estimator is adaptive over a class of Besov
spaces, namely, we prove that it achieves the same rates of convergence as in
the i.i.d framework
Signal detection for inverse problems in a multidimensional framework
International audienceThis paper is devoted to multi-dimensional inverse problems. In this setting, we address a goodness-of-fit testing problem. We investigate the separation rates associated to different kinds of smoothness assumptions and different degrees of ill-posedness
Learning near-optimal policies with Bellman-residual minimization based fitted policy iteration and a single sample path
We consider the problem of finding a near-optimal policy in continuous space, discounted Markovian Decision Problems given the trajectory of some behaviour policy. We study the policy iteration algorithm where in successive iterations the action-value functions of the intermediate policies are obtained by picking a function from some fixed function set (chosen by the user) that minimizes an unbiased finite-sample approximation to a novel loss function that upper-bounds the unmodified Bellman-residual criterion. The main result is a finite-sample, high-probability bound on the performance of the resulting policy that depends on the mixing rate of the trajectory, the capacity of the function set as measured by a novel capacity concept that we call the VC-crossing dimension, the approximation power of the function set and the discounted-average concentrability of the future-state distribution. To the best of our knowledge this is the first theoretical reinforcement learning result for off-policy control learning over continuous state-spaces using a single trajectory
Robust density estimation with the -loss. Applications to the estimation of a density on the line satisfying a shape constraint
We solve the problem of estimating the distribution of presumed i.i.d.
observations for the total variation loss. Our approach is based on density
models and is versatile enough to cope with many different ones, including some
density models for which the Maximum Likelihood Estimator (MLE for short) does
not exist. We mainly illustrate the properties of our estimator on models of
densities on the line that satisfy a shape constraint. We show that it
possesses some similar optimality properties, with regard to some global rates
of convergence, as the MLE does when it exists. It also enjoys some adaptation
properties with respect to some specific target densities in the model for
which our estimator is proven to converge at parametric rate. More important is
the fact that our estimator is robust, not only with respect to model
misspecification, but also to contamination, the presence of outliers among the
dataset and the equidistribution assumption. This means that the estimator
performs almost as well as if the data were i.i.d. with density in a
situation where these data are only independent and most of their marginals are
close enough in total variation to a distribution with density . We also
show that our estimator converges to the average density of the data, when this
density belongs to the model, even when none of the marginal densities belongs
to it. Our main result on the risk of the estimator takes the form of an
exponential deviation inequality which is non-asymptotic and involves explicit
numerical constants. We deduce from it several global rates of convergence,
including some bounds for the minimax -risks over the sets of
concave and log-concave densities. These bounds derive from some specific
results on the approximation of densities which are monotone, convex, concave
and log-concave. Such results may be of independent interest
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