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International audience; Additive varying coefficient models are a natural extension of multiple linear regression models, allowing the regression coefficients to be functions of other variables. Therefore these models are more flexible to model more complex dependencies in data structures. In this paper we consider the problem of selecting in an automatic way the significant variables among a large set of variables, when the interest is on a given response variable. In recent years several grouped regularization methods have been proposed and in this paper we present these under one unified framework in this varying coefficient model context. For each of the discussed grouped regularization methods we investigate the optimization problem to be solved, possible algorithms for doing so, and the variable and estimation consistency of the methods. We investigate the finite-sample performance of these methods, in a comparative study, and illustrate them on real data examples.

International audience; This article extends the nonnegative garrote method to a component selection method in a nonparametric additive model in which each univariate function is estimated with P-splines. We also establish the consistency of the procedure. An advantage of P-splines is that the fitted function is represented in a rather small basis of B-splines. A numerical study illustrates the finite-sample performance of the method and includes a comparison with other methods. The nonnegative garrote method with P-splines has the advantage of being computationally fast and performs, with an appropriate parameter selection procedure implemented, overall very well. Real data analysis leads to interesting findings. Supplementary materials for this article (technical proofs, additional numerical results, R code) are available online.

International audience; One of the popular method for fitting a regression function is regularization: minimizing an objective function which enforces a roughness penalty in addition to coherence with the data. This is the case when formulating penalized likelihood regression for exponential families. Most of the smoothing methods employ quadratic penalties, leading to linear estimates, and are in general incapable of recovering discontinuities or other important attributes in the regression function. In contrast, non-linear estimates are generally more accurate. In this paper, we focus on non-parametric penalized likelihood regression methods using splines and a variety of non-quadratic penalties, pointing out common basic principles. We present an asymptotic analysis of convergence rates that justifies the approach. We report on a simulation study including comparisons between our method and some existing ones. We illustrate our approach with an application to Poisson non-parametric regression modeling of frequency counts of reported acquired immune deficiency syndrome (AIDS) cases in the UK.

Special issue: Multiscale Methods and Statistics: A Productive Marriage; International audience; We consider a nonparametric noisy data model $Y_k=f(x_k)+\epsilon_k$, $k=1, \ldots, n$, where the unknown signal $f: \ [0,1] \rightarrow{\mathbb{R}}$ is assumed to belong to a wide range of function classes, including discontinuous functions, and the $\epsilon_k's$ are independent identically distributed noises with zero median. The distribution of the noise is assumed to be unknown and to satisfy some weak conditions. Possible noise distributions may have heavy tails, so that, for example, no moments of the noise exist. The design points are assumed to be deterministic points, not necessarily equispaced within the interval $[0, 1]$. Since the functions can be nonsmooth and the noise may have heavy tails, traditional estimation methods (for example, kernel methods) cannot be applied directly in this situation. As in Brown, Cai, and Zhou (2008), our approach first uses local medians to construct certain variables $Z_k$ structured as a Gaussian nonparametric regression but, unlike in this paper, the resulting data being not equispaced, we apply a wavelet block penalizing procedure adapted to non-equidistant designs to construct an estimator of the regression function. Under mild assumptions on the design it is shown that our estimator simultaneously attains the optimal rate of convergence over a wide range of Besov classes, without prior knowledge of the smoothness of the underlying functions or prior knowledge of the error distribution. The performance of our procedures is evaluated on simulated data sets covering a broad variety of settings and on some data examples, and are compared with other proposals made in the literature for treating similar problems.

In this paper we consider a regression model with errors that are martingale differences. This modeling includes the regression of both independent and time series data. The aim is to study the appearance of structural breaks in both the mean and the variance functions, assuming that such breaks may occur simultaneously in both the functions. We develop nonparametric testing procedures that simultaneously test for structural breaks in the conditional mean and the conditional variance. The asymptotic distribution of an adaptive test statistic is established, as well as its asymptotic consistency and efficiency. Simulations illustrate the performance of the adaptive testing procedure. An application to the analysis of financial time series also demonstrates the usefulness of the proposed adaptive test in practice.