Vincent Rivoirard

Lasso and probabilistic inequalities for multivariate point processes

Due to its low computational cost, Lasso is an attractive regularization method for high-dimensional statistical settings. In this paper, we consider multivariate counting processes depending on an unknown function to be estimated by linear combinations of a fixed dictionary. To select coefficients, we propose an adaptive

BERNSTEIN-VON MISES THEOREM FOR LINEAR FUNCTIONALS OF THE DENSITY

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Nonparametric Estimation of the Division Rate of a Size-Structured Population

-penalization methodology, where data-driven weights of the penalty are derived from new Bernstein type inequalities for martingales. Oracle inequalities are established under assumptions on the Gram matrix of the dictionary. Non-asymptotic probabilistic results for multivariate Hawkes processes are proven, which allows us to check these assumptions by considering general dictionaries based on histograms, Fourier or wavelet bases. Motivated by problems of neuronal activities inference, we finally lead a simulation study for multivariate Hawkes processes and compare our methodology with the {\it adaptive Lasso procedure} proposed by Zou in \cite{Zou}. We observe an excellent behavior of our procedure with respect to the problem of supports recovery. We rely on theoretical aspects for the essential question of tuning our methodology. Unlike adaptive Lasso of \cite{Zou}, our tuning procedure is proven to be robust with respect to all the parameters of the problem, revealing its potential for concrete purposes, in particular in neuroscience.

Near optimal thresholding estimation of a Poisson intensity on the real line

In this paper, we study the asymptotic posterior distribution of linear functionals of the density by deriving general conditions to obtain a semi-parametric version of the Bernstein–von Mises theorem. The special case of the cumulative distributive function, evaluated at a specific point, is widely considered. In particular, we show that for infinite-dimensional exponential families, under quite general assumptions, the asymptotic posterior distribution of the functional can be either Gaussian or a mixture of Gaussian distributions with different centering points. This illustrates the positive, but also the negative, phenomena that can occur in the study of Bernstein–von Mises results.

Adaptive Dantzig density estimation

We consider the problem of estimating thedivision rate of a size-structured population in anonparametric setting. The size of the system evolvesaccording to a transport-fragmentation equation: eachindividual grows with a given transport rate, and splitsinto two offsprings of the same size, followinga binary fragmentation process with unknown division ratethat depends on its size. In contrast to a deterministicinverse problem approach, we take in this talk theperspective of statistical inference: our data consists ina large sample of the size of individuals, when theevolution of the system is close to its time-asymptoticbehavior, so that it can be related to the eigenproblem ofthe considered transport-fragmentation equation. Byestimating statistically each term of the eigenvalueproblem and by suitably inverting a certain linearoperator, we are able to construct a more realisticestimator of the division rate that achieves the sameoptimal error bound as in related deterministic inverseproblems. Our procedure relies on kernel methods withautomatic bandwidth selection.

Posterior Concentration Rates for Infinite Dimensional Exponential Families

The purpose of this paper is to estimate the intensity of a Poisson process N by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of N with respect to ndx where n is a fixed parameter, is assumed to be non-compactly supported. The estimator ˜ fn,γ based on random thresholds is proved to achieve the same performance as the oracle estimator up to a possible logarithmic term. Then, minimax properties of ˜ fn,γ on Besov spaces B α p,q are established. Under mild assumptions, we prove that

Adaptive pointwise estimation of conditional density function

This paper deals with the problem of density estimation. We aim at building an estimate of an unknown density as a linear combination of functions of a dictionary. Inspired by Candes and Taos approach, we propose an

Posterior concentration rates for empirical Bayes procedures with applications to Dirichlet process mixtures

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LASSO-type estimators for semiparametric nonlinear mixed-effects models estimation

-minimization under an adaptive Dantzig constraint coming from sharp concentration inequalities. This allows to consider a wide class of dictionaries. Under local or global coherence assumptions, oracle inequalities are derived. These theoretical results are also proved to be valid for the natural Lasso estimate associated with our Dantzig procedure. Then, the issue of calibrating these procedures is studied from both theoretical and practical points of view. Finally, a numerical study shows the significant improvement obtained by our procedures when compared with other classical procedures.

Estimator selection: a new method with applications to kernel density estimation

In this paper we derive adaptive non-parametric rates of concentration of the posterior distributions for the density model on the class of Sobolev and Besov spaces. For this purpose, we build prior models based on wavelet or Fourier expansions of the logarithm of the density. The prior models are not necessarily Gaussian