Adaptive estimation of the stationary density of discrete and continuous time mixing processes

Fabienne Comte; Florence Merlevède

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In this paper, we study the problem of non parametric estimation of an unknown regression function from dependent data with sub-gaussian errors. As a particular case, we handle the autoregressive framework. For this purpose, we consider a collection of finite dimensional linear spaces (e.g. linear spaces spanned by wavelets or piecewise polynomials on a possibly irregular grid) and we estimate the regression function by a least-squares estimator built on a data driven selected linear...

Exact adaptive pointwise estimation on Sobolev classes of densities

Cristina Butucea (2001)

ESAIM: Probability and Statistics

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The subject of this paper is to estimate adaptively the common probability density of $n$ independent, identically distributed random variables. The estimation is done at a fixed point $x_{0} \in ℝ$ , over the density functions that belong to the Sobolev class $W_{n} (β, L)$ . We consider the adaptive problem setup, where the regularity parameter $β$ is unknown and varies in a given set $B_{n}$ . A sharp adaptive estimator is obtained, and the explicit asymptotical constant, associated to its rate of convergence is found. ...

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ESAIM: Probability and Statistics

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Nonparametric density estimators based on nonstationary absolutely regular random sequences.

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Convergence of weighted linear process for $ρ$ -mixing random variables.

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Adaptive estimation of a quadratic functional of a density by model selection

Béatrice Laurent (2005)

ESAIM: Probability and Statistics

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We consider the problem of estimating the integral of the square of a density $f$ from the observation of a $n$ sample. Our method to estimate $\int_{ℝ} f^{2} (x) d x$ is based on model selection via some penalized criterion. We prove that our estimator achieves the adaptive rates established by Efroimovich and Low on classes of smooth functions. A key point of the proof is an exponential inequality for $U$ -statistics of order 2 due to Houdré and Reynaud.

Detecting abrupt changes in random fields

Antoine Chambaz (2002)

ESAIM: Probability and Statistics

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This paper is devoted to the study of some asymptotic properties of a $M$ -estimator in a framework of detection of abrupt changes in random field’s distribution. This class of problems includes e.g. recovery of sets. It involves various techniques, including $M$ -estimation method, concentration inequalities, maximal inequalities for dependent random variables and $φ$ -mixing. Penalization of the criterion function when the size of the true model is unknown is performed. All the results apply...

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Minimax and bayes estimation in deconvolution problem

Mikhail Ermakov (2008)

ESAIM: Probability and Statistics

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We consider a deconvolution problem of estimating a signal blurred with a random noise. The noise is assumed to be a stationary Gaussian process multiplied by a weight function function where and is a small parameter. The underlying solution is assumed to be infinitely differentiable. For this model we find asymptotically minimax and Bayes estimators. In the case of solutions having finite number of derivatives similar results were obtained in [G.K. Golubev and R.Z. Khasminskii,...