Showing papers by "Jean-Michel Loubes published in 2011"
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TL;DR: Using barycenters of measures in the Wasserstein space, an iterative version as an estimation of the mean distribution is proposed, when the distributions are a common measure warped by a centered random operator, then the barycenter enables to recover this distribution template.
Abstract: In this paper we tackle the problem of comparing distributions of random variables and defining a mean pattern between a sample of random events. Using barycenters of measures in the Wasserstein space, we propose an iterative version as an estimation of the mean distribution. Moreover, when the distributions are a common measure warped by a centered random operator, then the barycenter enables to recover this distribution template.
61 citations
TL;DR: In this article, the authors consider ill-posed inverse problems and prove that the inversion of the operator is not always necessary and propose tests strategies attaining the necessary test rates.
34 citations
TL;DR: In this article, a nonparametric warping model for functional data is introduced, where data can be seen as realizations of a stochastic process, which takes into account the variations between the different observed curves.
Abstract: This article introduces a non parametric warping model for functional data. When the outcome of an experiment is a sample of curves, data can be seen as realizations of a stochastic process, which takes into account the variations between the different observed curves. The aim of this work is to define a mean pattern which represents the main behaviour of the set of all the realizations. So, we define the structural expectation of the underlying stochastic function. Then, we provide empirical estimators of this structural expectation and of each individual warping function. Consistency and asymptotic normality for such estimators are proved.
15 citations
TL;DR: In this paper, the covariance matrix of a stochastic process corrupted by additive noise is estimated in a high-dimensional setting under the assumption that the process has a sparse representation in a large dictionary.
Abstract: 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. Using such a penalty, the method selects a sparse set of basis functions in the dictionary used to approximate the process, leading to an approximation of the covariance matrix into a low dimensional space. Consistency of the estimator is studied in Frobenius and operator norms and an application to sparse PCA is proposed.
12 citations
TL;DR: In this article, the authors extend the calibration technique by using a maximum entropy method, which is achieved by considering random weights and looking for a discrete distribution which maximizes an entropy under the calibration constraint.
5 citations
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TL;DR: In this article, the problem of finding a good representative of a sample of random curves warped from a common pattern was moved onto a manifold framework, and an estimation of the common pattern $f$ based on an approximated geodesic distance on a suitable manifold was proposed.
Abstract: We focus on the problem of finding a good representative of a sample of random curves warped from a common pattern $f$. We first prove that such an problem can be moved onto a manifold framework. Then, we propose an estimation of the common pattern $f$ based on an approximated geodesic distance on a suitable manifold. We then compare the proposed method to more classical methods.
3 citations
TL;DR: In this paper, a projection operator built by plugging an empirical covariance estimator into a Schur complement decomposition of the projector is used to compute the blind prediction of a Gaussian time series.
Abstract: We tackle the issue of the blind prediction of a Gaussian time series. For this, we construct a projection operator built by plugging an empirical covariance estimator into a Schur complement decomposition of the projector. This operator is then used to compute the predictor. Rates of convergence of the estimates are given.
2 citations
TL;DR: In this article, the authors considered a wavelet series with an additive Brownian motion and established the local and asymptotic normality of the model, with respect to this two parameters.
Abstract: We consider a lacunar wavelet series function observed with an additive Brownian motion. Such functions are statistically characterized by two parameters. The first parameter governs the lacunarity of the wavelet coefficients while the second one governs its intensity. In this paper, we establish the local and asymptotic normality (LAN) of the model, with respect to this couple of parameters. This enables to prove the optimality of an estimator for the lacunarity parameter, and to build optimal (in the Le Cam sense) tests on the intensity parameter. © 2011 EDP Sciences, SMAI.
2 citations
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TL;DR: In this article, the authors give a general construction of stationary Gaussian processes indexed on graphs, which relies on spectral theory of Hilbertian operators defined on a graph, and extend natural maximum likelihood estimators of the parameters of the corresponding spectral density and provide their asymptotic behaviour.
Abstract: In this paper, we give a general construction of stationary Gaussian processes indexed on graphs. This construction relies on spectral theory of Hilbertian operators defined on a graph. We then extend natural maximum likelihood estimators of the parameters of the corresponding spectral density and provide their asymptotic behaviour.
1 citations
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TL;DR: In this article, the authors consider a model selection estimator of the covariance of a random process using the Unbiased Risk Estimation (URE) method, which allows to select an estimator in a collection of model.
Abstract: We consider a model selection estimator of the covariance of a random process. Using the Unbiased Risk Estimation (URE) method, we build an estimator of the risk which allows to select an estimator in a collection of model. Then, we present an oracle inequality which ensures that the risk of the selected estimator is close to the risk of the oracle. Simulations show the efficiency of this methodology.
1 citations
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TL;DR: In this article, asymptotic expansion of the likelihood of a certain class of Gaussian processes characterized by their spectral density was studied and proved LAN property for these models which include in particular fractional Brownian motion %$B^\alpha_t,\: \alpha \geq 1/2$ or ARFIMA processes.
Abstract: We study asymptotic expansion of the likelihood of a certain class of Gaussian processes characterized by their spectral density $f_\theta$. We consider the case where $f_\theta\PAR{x} \sim_{x\to 0} \ABS{x}^{-\al(\theta)}L_\theta(x)$ with $L_\theta$ a slowly varying function and $\al\PAR{\theta}\in (-\infty,1)$. We prove LAN property for these models which include in particular fractional Brownian motion %$B^\alpha_t,\: \alpha \geq 1/2$ or ARFIMA processes.
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TL;DR: In this paper, an estimator of the conditional covariance matrix in an inverse regression setting based on the estimation of a quadratic functional was proposed, which provides an efficient estimator from a semi parametric point of view.
Abstract: Let $\boldsymbol{X}\in \mathbb{R}^p$ and $Y\in \mathbb{R}$ In this paper we propose an estimator of the conditional covariance matrix, $\mathrm{Cov}(\mathbb{E}[\boldsymbol{X}\vert Y])$, in an inverse regression setting Based on the estimation of a quadratic functional, this methodology provides an efficient estimator from a semi parametric point of view We consider a functional Taylor expansion of $\mathrm{Cov}(\mathbb{E}[\boldsymbol{X}\vert Y])$ under some mild conditions and the effect of using an estimate of the unknown joint distribution The asymptotic properties of this estimator are also provided