Section: New Results

Sparsity and ${\ell }_1$–regularization

Participant : Vincent Rivoirard.

For multivariate Hawkes processes

Motivated by statistical problems in neuroscience, Vincent Rivoirard and his coauthors study in [31] nonparametric inference for multivariate Hawkes processes depending on an unknown function to be estimated by linear combinations of a fixed dictionary. To select coefficients, they propose a Lasso-type 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 are proven, which allows them to check these assumptions by considering general dictionaries based on histograms, Fourier or wavelet bases. They finally carry out a simulation study and compare their methodology with the adaptive Lasso procedure proposed by Zou. They observe an excellent behavior of their procedure with respect to the problem of supports recovery. Unlike adaptive Lasso of Zou, their tuning procedure is proven to be robust with respect to all the parameters of the problem, revealing its potential for concrete purposes in neuroscience, but also in other fields.

In the spherical convolution model

In [21] , Thanh Mai Pham Ngoc and Vincent Rivoirard consider the problem of estimating a density of probability from indirect data in the spherical convolution model. They aim at building an estimate of the unknown density as a linear combination of functions of an overcomplete dictionary. The procedure is devised through a well-calibrated ${\ell }_1$–penalized criterion. The dictionary approach allows to combine various bases and thus enhances estimates sparsity. They provide an oracle inequality under global coherence assumptions. Moreover, the calibrated procedure that they put forward gives very satisfactory results in the numerical study when compared with other procedures.

For semiparametric nonlinear mixed-effects models

Semiparametric nonlinear mixed-effects models (SNMMs) have been proposed as an extension of nonlinear mixed-effects models (NLMMs). These models are a good compromise and retain nice features of both parametric and nonparametric models resulting in more flexible models than standard parametric NLMMs. In [28] , Vincent Rivoirard and his coauthors propose new estimation strategies in SNMMs. They propose a Lasso-type method to estimate the unknown nonlinear function. They derive oracle inequalities for this nonparametric estimator. They combine the two approaches in a general estimation procedure that they illustrate with simulations and through the analysis of a real data set of price evolution in on-line auctions.