We propose an adaptive Lasso estimator for inferring the fertility function of a Hawkes stationary point process, without any assumption on its form. We study both the numerical and theoretical properties of this estimator. We will show its numerical advantages over the classical CLS estimator and over the non-adaptive Lasso estimator, both in simulation and when applied to a set of epidemiological data. The theoretical properties of this estimator will instead be obtained by embedding this problem in the more general one of estimating the coefficients of an AR(p) model with stationary noise. This last result has its own specific interest because it enlarges the applicability of the adaptive LASSO procedure to the case of more general time series than those usually treated in the literature
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