Motivated by the analysis of a Positron Emission Tomography (PET) imaging data considered in Bowen et al. (2012), we introduce a semiparametric topographical mixture model able to capture the characteristics of dichotomous shifted response-type experiments. We propose a local estimation procedure, based on the symmetry of the local noise, for the proportion and locations functions involved in the proposed model.
We establish under mild conditions the minimax properties and asymptotic normality of our estimators when Monte Carlo simulations are conducted to examine their nite sample performance. Finally a statistical analysis of the PET imaging data in Bowen et al. (2012) is illustrated for the proposed method.

We consider copulas with a given diagonal section and compute the explicit density of the unique optimal copula which maximizes the entropy. The entropy can also be written as the relative entropy with respect to the independent copula. In this sense, this copula is the least informative one among the copulas with a given diagonal section. We give an explicit criterion on the diagonal section for the existence of the optimal copula and give a closed formula for its relative entropy. We also provide examples for some diagonal sections of usual bivariate copulas and illustrate the differences between them and the maximum entropy copula with the same diagonal section. An interesting application of the results concerning copulas of order statistics will also be presented.

This paper deals with the trace regression model where n entries or linear combinations of entries of an unknown m1×m2 matrix A0 corrupted by noise are observed. We propose a new nuclear norm penalized estimator of A0 and establish a general sharp oracle inequality for this estimator for arbitrary values of n,m1,m2 under the condition of isometry in expectation. Then this method is applied to the matrix completion problem. In this case, the estimator admits a simple explicit form and we prove that it satisfies oracle inequalities with faster rates of convergence than in the previous works. They are valid, in particular, in the high-dimensional setting m1m2≫n. We show that the obtained rates are optimal up to logarithmic factors in a minimax sense and also derive, for any fixed matrix A0, a non-minimax lower bound on the rate of convergence of our estimator, which coincides with the upper bound up to a constant factor. Finally, we show that our procedure provides an exact recovery of the rank of A0 with probability close to 1. We also discuss the statistical learning setting where there is no underlying model determined by A0 and the aim is to find the best trace regression model approximating the data.

(Joint work with Vladimir Koltchinskii and Alexander B. Tsybakov)

After a short introduction of the notion of Ricci curvature in the continuous setting, I will present a notion of an interpolating path, on graph, and prove some displacement convexity of the entropy along such an interpolation. If time allows I will give some application to functional inequalities of Poincaré, log-Sobolev and transport-entropy type.

Joint work with N. Gozlan, P.M. Samson and P. Tetali (http://perso-math.univ-mlv.fr/users/roberto.cyril/articles/displacement-...).

Consider two large random matrix models of interest in the applications: the large covariance matrix and the signal plus noise model. In this talk, we will briefly describe the fluctuations of the linear spectral statistics associated to these models. Often in large random matrix theory, the normalized trace of the resolvent is an efficient device to describe the spectrum and its fluctuations. The central part of the talk will be devoted to the presentation of a method which enables to recover the fluctuations for functions with low regularity (say 3 times differentiable functions) from the fluctuations of the trace of the resolvent. We will rely on Helffer-Sjostrand formula to represent the linear statistics of interest and on recent variance estimates obtained by Shcherbina.

Cet exposé présentera un panorama partiel et quelques résultatsrécents sur le comportement asymptotique (moyenne, variance, lois limites) dans certains problèmes de plusl ongue sous-suite croissante et/ou commune.

L'estimation par minimisation du contraste des moindres carrés dans un modèle de régression linéaire est en général bien connue de tous. Lorsque la dépendance entre les variables n'est plus considérée comme a priori linéaire, des stratégies non paramétriques visant à estimer une fonction quelconque ont été mises en place. Cet exposé a pour but de présenter les méthodes statistiques actuelles utilisées pour définir et étudier des estimateurs fonctionnels fondés sur des contrastes de types moindres carrés et des techniques de sélection de modèle par pénalisation. Les outils d'analyse et de probabilité utiles aux travaux statistiques seront mis en évidence. Les fonctions considérées sont tout d'abord les classiques densité et fonction de régression, mais on envisagera ensuite la densité conditionnelle, ou le risque instantané (hazard rate) dans les modèles de survie, en présence ou non de censure ou de conditionnement.

Le processus TCP est utilisé pour gérer les congestions lors de l'envoi de données sur internet. Nous étudions une modélisation simple de ce processus, pour laquelle nous recherchons des estimées pour les vitesses de convergence à l'équilibre en distance de Wasserstein et en variation totale. Les caractères non-réversible et "peu aléatoire" de ce processus nous obligent à développer de nouvelles méthodes de couplage, qui peuvent aussi s'avérer utiles pour d'autres processus de Markov. (travail en commun avec A. Christen, A. Guillin, F. Malrieu et P.-A. Zitt.)