Florent Autin

Thresholding methods to estimate copula density

This paper deals with the problem of multivariate copula density estimation. Using wavelet methods we provide two shrinkage procedures based on thresholding rules for which knowledge of the regularity of the copula density to be estimated is not necessary. These methods, said to be adaptive, have proved to be very effective when adopting the minimax and the maxiset approaches. Moreover we show that these procedures can be discriminated in the maxiset sense. We provide an estimation algorithm and evaluate its properties using simulation. Finally, we propose a real life application for financial data.

Test on components of mixture densities

Abstract This 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 Y1,...,Yn and Z1,...,Zn 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; this link had never been clearly established up to now.

Ideal denoising within a family of tree-structured wavelet estimators

We focus on the performances of tree-structured wavelet estimators belonging to a large family of keep-or-kill rules, namely the Vertical Block Thresholding family. For each estimator, we provide the maximal functional space (maxiset) for which the quadratic risk reaches a given rate of convergence. Following a discussion on the maxiset embeddings, we identify the ideal estimator of this family, that is the one associated with the largest maxiset. We emphasize the importance of such a result since the ideal estimator is different from the usual (plug-in) estimator used to mimic the performances of the Oracle. Finally, we confirm the good performances of the ideal estimator compared to the other elements of that family through extensive numerical experiments.

On the performances of a new thresholding procedure using tree structure

This paper deals with the problem of function estimation. Us- ing 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.

Combining thresholding rules: a new way to improve the performance of wavelet estimators

In this paper, we address the situation where we cannot differentiate wavelet-based threshold procedures because their sets of well-estimated functions (maxisets) are not nested. As a generic solution, we propose to proceed via a combination of these procedures in order to achieve new procedures which perform better in the sense that the involved maxisets contain the union of the previous ones. Throughout the paper we propose illuminating interpretations of the maxiset results and provide conditions to ensure that this combination generates larger maxisets. As an example, we propose to combine vertical- and horizontal-block thresholding procedures that are already known to perform well. We discuss the limitation of our method, and we check our theoretical results through numerical experiments.

Asymptotic Performance of Projection Estimators in Standard and Hyperbolic Wavelet Bases

We provide a novel treatment of the ability of the standard (wavelet-tensor) and of the hyperbolic (tensor product) wavelet bases to build nonparametric estimators of multivariate functions. First, we give new results about the limitations of wavelet estimators based on the standard wavelet basis regarding their inability to optimally reconstruct functions with anisotropic smoothness. Next, we provide optimal or near optimal rates at which both linear and non-linear hyperbolic wavelet estimators are well-suited to reconstruct functions from anisotropic Besov spaces and subsequently we characterize the set of all the functions that are well reconstructed by these methods with respect to these rates. As a first main result, we furnish novel arguments to understand the primordial role of sparsity and thresholding in multivariate contexts, in particular by showing a stronger exposure of linear methods to the curse of dimensionality. Second, we propose an adaptation of the well known block thresholding method to a hyperbolic wavelet basis and show its ability to estimate functions with anisotropic smoothness at the optimal minimax rate. Therefore, we prove the pertinence of horizontal information pooling even in high dimensional settings. Numerical experiments illustrate the finite samples properties of the studied estimators.

Adaptive test on components of densities mixture

AbstractWe are interested in testing whether two independent samples of n independent random variables are based on the same mixing-components or not. We provide a test procedure to detect if at least two mixing-components are distinct when their difference is a smooth function. Our test procedure is proved to be optimal according to the minimax adaptive setting that differs from the minimax setting by the fact that the smoothness is not known. Moreover, we show that the adaptive minimax rate suffers from a loss of order (

Large variance Gaussian priors in Bayesian nonparametric estimation: a maxiset approach

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