Robust Estimation and Wavelet Thresholding in Partial Linear Models

This paper is concerned with a semiparametric partially linear regression model with unknown regression coefficients, an unknown nonparametric function for the non-linear component, and unobservable Gaussian distributed random errors. We present a wavelet thresholding based estimation procedure to estimate the components of the partial linear model by establishing a connection between an $l_1$-penalty based wavelet estimator of the nonparametric component and Huber's M-estimation of a standard linear model with outliers. Some general results on the large sample properties of the estimates of both the parametric and the nonparametric part of the model are established. Simulations and a real example are used to illustrate the general results and to compare the proposed methodology with other methods available in the recent literature

Classification of EEG recordings in auditory brain activity via a logistic functional linear regression model

We want to analyse EEG recordings in order to investigate the phonemic categorization at a very early stage of auditory processing. This problem can be modelled by a supervised classification of functional data. Discrimination is explored via a logistic functional linear model, using a wavelet representation of the data. Different procedures are investigated, based on penalized likelihood and principal component reduction or partial least squares reduction

Estimation par ondelettes dans des modèles fonctionnels généralisés

National audienceL'estimation par ondelettes de signaux en présence de bruit gaussien a été largement développée ces dernières années. Le but de ce travail est d'étendre les résultats à des contextes faisant appel à des distributions plus générales telles que Poisson, Binomiale ou Gamma... Nous considérons une approche par log-vraisemblance pénalisée, où la pénalité s'exprime à l'aide des coefficients d'ondelettes. Nous nous intéressons par ailleurs au cas où un terme linéaire est estimé simultanément. Nous montrons l'optimalité asymptotique de la procédure d'estimation et nous proposons un algorithme simple de mise en oeuvre

Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave

Multivariate time series with long-dependence are observed in many applications such as finance, geophysics or neuroscience. Many packages provide estimation tools for univariate settings but few are addressing the problem of long-dependence estimation for multivariate settings. The package multiwave is providing efficient estimation procedures for multivariate time series. Two semi-parametric estimation methods of the long-memory exponents and long-run covariance matrix of time series are implemented. The first one is the Fourier-based estimation proposed by Shimotsu (2007) and the second one is a wavelet-based estimation described in Achard and Gannaz (2016). The objective of this paper is to provide an overview of the R package multiwave with its practical application perspectives

Adaptive density estimation under dependence

Assume that $(X_t)_{t\in\Z}$ is a real valued time series admitting a common marginal density $f$ with respect to Lebesgue's measure. Donoho {\it et al.} (1996) propose a near-minimax method based on thresholding wavelets to estimate $f$ on a compact set in an independent and identically distributed setting. The aim of the present work is to extend these results to general weak dependent contexts. Weak dependence assumptions are expressed as decreasing bounds of covariance terms and are detailed for different examples. The threshold levels in estimators $\widehat f_n$ depend on weak dependence properties of the sequence $(X_t)_{t\in\Z}$ through the constant. If these properties are unknown, we propose cross-validation procedures to get new estimators. These procedures are illustrated via simulations of dynamical systems and non causal infinite moving averages. We also discuss the efficiency of our estimators with respect to the decrease of covariances bounds

Wavelet penalized likelihood estimation in generalized functional models

The paper deals with generalized functional regression. The aim is to estimate the influence of covariates on observations, drawn from an exponential distribution. The link considered has a semiparametric expression: if we are interested in a functional influence of some covariates, we authorize others to be modeled linearly. We thus consider a generalized partially linear regression model with unknown regression coefficients and an unknown nonparametric function. We present a maximum penalized likelihood procedure to estimate the components of the model introducing penalty based wavelet estimators. Asymptotic rates of the estimates of both the parametric and the nonparametric part of the model are given and quasi-minimax optimality is obtained under usual conditions in literature. We establish in particular that the LASSO penalty leads to an adaptive estimation with respect to the regularity of the estimated function. An algorithm based on backfitting and Fisher-scoring is also proposed for implementation. Simulations are used to illustrate the finite sample behaviour, including a comparison with kernel and splines based methods

Ondelettes et modèles partiellement lineaires généralisés.

National audienceLes modèles partiellement linéaires distinguent dans un signal des relations linéaires et des relations fonctionnelles, non paramétriques

Estimation par ondelettes dans les modèles partiellement linéaires

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Inference of dependence graphs by multiple testing, with application to brain connectivity

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