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In this article we study the asymptotic behaviour of the least square estimator in a linear regression model based on random observation instances. We provide mild assumptions on the moments and dependence structure on the randomly spaced observations and the residuals under which the estimator is strongly consistent. In particular, we consider observation instances that are negatively superadditive dependent within each other, while for the residuals we merely assume that they are generated by some continuous function. In addition, we prove that the rate of convergence is proportional to the sampling rate $N$, and we complement our findings with a simulation study providing insights on finite sample properties., Final version will be published in Statistics

International audience; We consider the problem of estimating a high-dimensional additive mixed model for longitudinal data using sparse methods. In this problem, multiple measurements are made on the same subject across time, and then the different sources of variability (intra- and inter-subject variability) and correlation within subjects have to be considered. Also, the relationships between explanatory variables and the outcome arepossibly non linear. In addition, the number of explanatory variables could be larger than the sample size but only a small set of explanatory variables contribute to the response.Several computational approaches for high-dimensional additive modelling for independent data have been developed in the literature. Recently, Amato and colleagues (Stat Methods Appl 2016; s10260-016-0357-8) conducted a comprehensive review of these methods. Efficient regularized estimation procedures for variable selection in nonparametric additive models use basis function approximations. The authors also proposed a reformulation of the estimation problem in terms of group Lasso that allows deducing convergence and asymptotic optimality properties.Only a few works have developed suggestions to analyse high-dimensional longitudinal data using Lasso-type methods in additive mixed model. The resulting estimator depends only on a relatively small number of basis functions, however variable selection is not directly encouraged. In this study we explore the extension of the group Lasso penalty to additive mixed effects models. We discuss computational aspects, including a comparison of group Lasso algorithms implemented through publicly available R codes, the estimation of optimal regularization parameter and linkages between mean and covariance parameter estimation algorithms. We illustrate the interest of such approaches in the analysis of a twenty - year longitudinal study of training practices of elite athletes.

International audience; The penalization of likelihoods by L1–norms has become a relatively standard technique for highdimensional data when the assumed models are based on n independent and identically distributed observations. These techniques should improve prediction accuracy (since regularization leads to variance reduction) together with interpretability (since sparsity identifies a subset of variables with strong effects). Computationally, these penalties are attractive and their theoretical properties have been intensively studied during the last years.Several authors have recently suggested analyzing high-dimensional clustered or longitudinal data using L1–penalization methods in mixed effects models. These approaches are mostly developed for variable selection purposes in linear and generalized linear mixed effects models and also, but less extensive, in parametric nonlinear mixed effects models. Only a few works have considered the problem of selecting nonlinear functions using L1–penalization methods in nonparametric mixed effects models, with additive or nonadditive predictors. Nonlinear functions are approximated by a linear combination of smooth functions (spline, wavelet or Fourier basis functions) possibly combined with more irregular functions (spiky basis functions). The resulting estimator depends only on a relatively small number of variables and/ora relatively small number of basis functions [1].In this study we illustrate the interest of such approaches in the analysis of the DALIA-1 longitudinal trial [2]. Eighteen HIV infected patients received vaccine injections at weeks 0, 4, 8 and 12. Antiretroviral treatment was interrupted at week 24. The patients were followed up to week 48, leading to 14 repeated measures per subject. Our aim was to predict the evolution of viral loads (continuous response) from the about 260 gene sets (predictors). The incorporation of the temporal effect is a key point to reach accurate predictions.References[1] Arribas-Gil, A. and Bertin, K. and Meza, C. and Rivoirard, V. (2012). LASSO-type estimators for Semiparametric Nonlinar Mixed-Effects Models Estimation, Statistics and Computing, 24 (3), 443-460.[2] Lévy, Y. and Thiébaut, R. and Montes, M. and Lacabaratz, C. and Sloan, L. and King, B. et al. (2014). Dendritic cell-based therapeutic vaccine elicits polyfunctional HIV-specific T-cell immunity associated with control of viral load. European journal of immunology, 44 (9), 2802-2810.

We propose a Bayesian approach to detect multiple change-points in a piecewise-constant signal corrupted by a functional part corresponding to environmental or experimental disturbances. The piecewise constant part (also called segmentation part) is expressed as the product of a lower triangular matrix by a sparse vector. The functional part is a linear combination of functions from a large dictionary. A Stochastic Search Variable Selection approach is used to obtain sparse estimations of the segmentation parameters (the change-points and the means over the segments) and of the functional part. The performance of our proposed method is assessed using simulation experiments. Applications to two real datasets from geodesy and economy fields are also presented., 36 pages, 7 figures

In this paper, our aim is to estimate an anisotropic function in the framework of the Gaussian white noise model. We suppose that the function belongs to an unknown d-dimensional Holder class. We evaluate the performance of an estimator by the sup-norm risk. We find the exact asymptotics of the adaptive minimax risk and construct asymptotically adaptive estimators.

In this paper, we are interested in the study of beta kernel estimators from an asymptotic minimax point of view. It is well known that beta kernel estimators are, on the contrary of classical kernel estimators, "free of boundary effect" and thus are very useful in practice. The goal of this paper is to prove that there is a price to pay: for very regular functions or for certain losses, these estimators are not minimax. Nevertheless they are minimax for classical regularities such as regularity of order two or less than two, supposed commonly in the practice and for some classical losses., 19 pages, 2 figures

FONDECYT 17 FONDECYT

We consider the Gaussian White Noise Model and we study the estimation of a function f in the uniform norm assuming that f belongs to a Holder anisotropic class. We give the minimax rate of convergence over this class and we determine the minimax exact constant and an asymptotically exact estimator.