Laboratoire de Probabilités, Statistique et Modélisation
Présentation
Le Laboratoire de Probabilités, Statistique et Modélisation, dans sa forme actuelle, a résulté, au 1er janvier 1999, de la fusion de l'ancien Laboratoire de probabilité de l'université Paris 6 avec l'équipe de Probabilités et statistique de l'université Paris Diderot.
Le laboratoire compte environ 70 enseignants-chercheurs permanents, 50 thésards, une équipe administrative de 6 personnes. Il accueille de plus les activités de deux masters deuxième année, ce qui représente plus de 200 étudiants chaque année.
La thématique du laboratoire s'inscrit dans le domaine des mathématiques appliquées et a pour objet la modélisation, la description et l'estimation des phénomènes aléatoires. Les thèmes de recherche abordés ici concernent des domaines très variés et recouvrent aussi bien des mathématiques fondamentales que des applications dans des domaines aussi divers que la médecine, les sciences humaines, l'astrophysique, les assurances ou la finance...
Thèmes de recherche
1. Théorie ergodique et systèmes dynamiques
2. Modélisation stochastique
3. Mouvement brownien et calcul stochastique
4. Statistiques
Equipes de recherche
Le laboratoire comprend six équipes :
- Théorie ergodique et systèmes dynamiques,
- Modélisation stochastique,
- Mouvement brownien et calcul stochastique,
- Statistique,
- Probabilités numériques et mathématiques financières,
- Probabilités-statistiques-biologie.
[hal-02107111] Second-order properties of regeneration-based bootstrap for atomic Markov chains
Date: 23 4 月 2019 - 16:14
Desc: In this paper, we show how the original bootstrap method introduced by Datta and McCormick (Can J Stat 21(2):181–193, 1993b), namely the regeneration-based bootstrap, for approximating the sampling distribution of sample mean statistics in the atomic Markovian setting can be modified to get the second-order accuracy. We prove that the drawback of the original construction mainly relies on an inaccurate estimation of the skewness of the sampling distribution by the bootstrap distribution and that it is possible to correct it by standardizing the regeneration-based bootstrap statistic by the length of the bootstrap series instead of the length of the sample and recentering the bootstrap distribution. An asymptotic result establishing the second-order accuracy of this bootstrap estimate up to O(n −1log (n)) (close to the rate obtained in the i.i.d. setting) is also stated under weak moment assumptions.
[hal-02107086] From Ranking to Classification: A Statistical View
Date: 23 4 月 2019 - 16:10
Desc: In applications related to information retrieval, the goal is not only to build a classifier for deciding whether a document x among a list χ is relevant or not, but to learn a scoring function s : χ → ℝ for ranking all possible documents with respect to their relevancy. Here we show how the bipartite ranking problem boils down to binary classification with dependent data when accuracy is measured by the A U C criterion. The natural estimate of the risk being of the form of a U-statistic, consistency of methods based on empirical risk minimization is studied using the theory of U-processes. Taking advantage of this specific form, we prove that fast rates of convergence may be achieved under general noise assumptions.
[hal-02107125] Regenerative block bootstrap for Markov chains
Date: 23 4 月 2019 - 16:16
Desc: A specific bootstrap method is introduced for positive recurrent Markov chains, based on the regenerative method and the Nummelin splitting technique. This construction involves generating a sequence of approximate pseudo-renewal times for a Harris chain X from data X1,..., Xn and the parameters of a minorization condition satisfied by its transition probability kernel and then applying a variant of the methodology proposed by Datta and McCormick for bootstrapping additive functionals of type n-1∑i=1nf(Xi) when the chain possesses an atom. This novel methodology mainly consists in dividing the sample path of the chain into data blocks corresponding to the successive visits to the atom and resampling the blocks until the (random) length of the reconstructed trajectory is at least n, so as to mimic the renewal structure of the chain. In the atomic case we prove that our method inherits the accuracy of the bootstrap in the independent and identically distributed case up to OP(n-1) under weak conditions. In the general (not necessarily stationary) case asymptotic validity for this resampling procedure is established, provided that a consistent estimator of the transition kernel may be computed. The second-order validity is obtained in the stationary case (up to a rate close to OP(n-1) for regular stationary chains). A data-driven method for choosing the parameters of the minorization condition is proposed and applications to specific Markovian models are discussed.
[hal-02107287] Sharp Bounds for the Tails of Functionals of Markov Chains
Date: 23 4 月 2019 - 16:57
Desc: This paper is devoted to establishing sharp bounds for deviation probabilities of partial sums Σ n i=1 f(Xi), where X = (Xn)nEN is a positive recurrent Markov chain and f is a real valued function defined on its state space. Combining the regenerative method to the Esscher transformation, these estimates are shown in particular to generalize probability inequalities proved in the i.i.d. case to the Markovian setting for (not necessarily uniformly) geometrically ergodic chains.
[hal-00688241] Target identification using dictionary matching of generalized polarization tensors
Date: 3 9 月 2024 - 22:33
Desc: The aim of this paper is to provide a fast and efficient procedure for (real-time) target identification in imaging based on matching on a dictionary of precomputed generalized polarization tensors (GPTs). The approach is based on some important properties of the GPTs and new invariants. A new shape representation is given and numerically tested in the presence of measurement noise. The stability and resolution of the proposed identification algorithm is numerically quantified. We compare the proposed GPT-based shape representation with a moment-based one.
Autres contacts
U.F.R. Mathématiques
Sophie-Germain
75013 PARIS
