Semiparametric Gaussian white noise models in the non-regular case ## AbstractThis paper deals with the behavior of some nonlinear semiparametric models in the case where the unknown nuisance function $f$ is not necessarily differentiable. Two models are considered, in the Gaussian white noise framework: estimation of the center of symmetry and estimation of the period of a periodic signal. We obtain the rate of convergence of the sieve maximum likelihood estimators in these models over different functional spaces. In particular, it is shown that if the class controls appropriately the growth to infinity of the Fisher information over the sieve, semiparametric fast rates of convergence are obtained. We also prove a lower bound result which implies that these semiparametric rates are strictly below the parametric ones, meaning there is a significant loss of information, contrary to the regular case.
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