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Estimation Lasso pour les processus de Hawkes multivariés

Motivated by statistical problems in neuroscience, we study nonparametric inference for multivariate Hawkes processes depending on an unknown function to be estimated by linear combinations of a fixed dictionary. To select coefficients, we 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 us to check these assumptions by considering general dictionaries based on histograms, Fourier or wavelet bases. We finally lead a simulation study and compare our methodology with the adaptive Lasso procedure proposed by Zou. We observe an excellent behavior of our procedure with respect to the problem of supports recovery. Unlike adaptive Lasso of 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 neuroscience, but also in other fields.

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