On the performances of a new thresholding procedure using tree structure

This paper deals with the problem of function estimation. Using the white noise model setting, we provide a method to construct a new wavelet procedure based on thresholding rules which takes advantage of the dyadic structure of the wavelet decomposition. We prove that this new procedure performs very well since, on the one hand, it is adaptive and near-minimax over a large class of Besov spaces and, on the other hand, the maximal functional space (maxiset) where this procedure attains a given rate of convergence is very large. More than this, by studying the shape of its maxiset, we prove that the new procedure outperforms the hard thresholding procedure.Comment: Published in at http://dx.doi.org/10.1214/08-EJS205 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Thresholding methods to estimate the copula density

This paper deals with the problem of the multivariate copula density estimation. Using wavelet methods we provide two shrinkage procedures based on thresholding rules for which the knowledge of the regularity of the copula density to be estimated is not necessary. These methods, said to be adaptive, are proved to perform very well when adopting the minimax and the maxiset approaches. Moreover we show that these procedures can be discriminated in the maxiset sense. We produce an estimation algorithm whose qualities are evaluated thanks some simulation. Last, we propose a real life application for financial data

Thresholding methods to estimate the copula density

This paper deals with the problem of the multivariate copula density estimation. Using wavelet methods we provide two shrinkage procedures based on thresholding rules for which the knowledge of the regularity of the copula density to be estimated is not necessary. These methods, said to be adaptive, are proved to perform very well when adopting the minimax and the maxiset approaches. Moreover we show that these procedures can be discriminated in the maxiset sense. We produce an estimation algorithm whose qualities are evaluated thanks some simulation. Last, we propose a real life application for financial data

Maxiset comparisons of procedures, application to choosing priors in a Bayesian nonparametric setting

In this paper our aim is to provide tools for easily calculating the maxisets of several procedures. Then we apply these results to perform a comparison between several Bayesian estimators in a non parametric setting. We obtain that many Bayesian rules can be described through a general behavior such as being shrinkage rules, limited, and/or elitist rules. This has consequences on their maxisets which happen to be automatically included in some Besov or weak Besov spaces, whereas other properties such as cautiousness imply that their maxiset conversely contains some of the spaces quoted above. We compare Bayesian rules taking into account the sparsity of the signal with priors which are combination of a Dirac with a standard distribution. We consider the case of Gaussian and heavy tail priors. We prove that the heavy tail assumption is not necessary to attain maxisets equivalent to the thresholding methods. Finally we provide methods using the tree structure of the dyadic aspect of the multiscale analysis, and related to Lepki's procedure, achieving strictly larger maxisets than those of thresholding methods

Maxiset point of view for signal detection in inverse problems

This paper extends the successful maxiset paradigm from function estimation to signal detection in inverse problems. In this context, the maxisets do not have the same shape compared to the classical estimation framework. Nevertheless, we introduce a robustified version of these maxisets, allowing to exhibit tail conditions on the signals of interest. Under this novel paradigm we are able to compare direct and indirect testing procedures

Test on components of mixture densities

International audienceThis paper deals with statistical tests on the components of mixture densities. We propose to test whether the densities of two independent samples of independent random variables $Y_1, \dots, Y_n$ and $Z_1, \dots, Z_n$ result from the same mixture of $M$ components or not. We provide a test procedure which is proved to be asymptotically optimal according to the minimax setting. We extensively discuss the connection between the mixing weights and the performance of the testing procedure and illustrate it with numerical examples

Point de vue maxiset en estimation non paramétrique

In the framework of a wavelet analysis, we study the statistical meaning of many classes of procedures. More precisely, we aim at investigating the maximal spaces (maxisets) where these procedures attain a given rate of convergence. The maxiset approach allows to bring theoretical explanations on some phenomena observed in the practical setting which are not explained by the minimax approach. Indeed, we show that data-driven thresholding rules outperform non random thresholding rules. Then, we prove that procedures which consist in thresholding coefficients by groups, as tree rules (close to Lepski's rule) or block thresholding rules, are often better in the maxiset sense than procedures which consist in thresholding coefficients individually. Otherwise, as many Bayesian rules built on heavy tailed densities, classical Bayesian rules built on Gaussian densities with large variance are proved to have maxisets which coincide with hard thresholding rules ones and to have very good numerical performances.Dans le cadre d'une analyse par ondelettes, nous étudions les propriétés statistiques de diverses classes de procédures. Plus précisément, nous cherchons à déterminer les espaces maximaux (maxisets) sur lesquels ces procédures atteignent une vitesse de convergence donnée. L'approche maxiset nous permet alors de donner une explication théorique à certains phénomènes observés en pratique et non expliqués par l'approche minimax. Nous montrons en effet que les estimateurs de seuillage aléatoire sont plus performants que ceux de seuillage déterministe. Ensuite, nous prouvons que les procédures de seuillage par groupes, comme certaines procédures d'arbre (proches de la procédure de Lepski) ou de seuillage par blocs, ont de meilleures performances au sens maxiset que les procédures de seuillage individuel. Par ailleurs, si les maxisets des estimateurs Bayésiens usuels construits sur des densités à queues lourdes sont de même nature que ceux des estimateurs de seuillage dur, nous montrons qu'il en est de même pour ceux des estimateurs Bayésiens construits à partir de densités Gaussiennes à grande variance et dont les performances numériques sont très bonnes