Second order properties in semiparametrics and applications
PhD thesis, Universite Paris-Sud.
This thesis is devoted to estimation in semiparametric models. We are interested in estimation in the Gaussian white noise model
$dX(t)=f(t/\theta)dt+dW(t),t\in[-T/2,T/2]$ where $f$ is an unknown 1-periodic function, $\theta$ an unknown parameter and $W(t)$ a standard Brownian motion. We obtain an asymptotic expansion at the order 2 for the quadratic risk of estimators of $\theta$ and find the best minimax speed for the second order term. This term is nothing but a nonparametric estimation speed. Then we consider the problem of nonparametric estimation of the function $f$ in this model, $\theta$ being the nuisance parameter. We use the fact that there are preliminary estimators of $\theta$ and build an estimator of $f$ which achieves the lower bound for the minimax quadratic risk in this problem. We illustrate our method by simulations and compare our algorithm to existing methods not taking into account the periodicity assumption. The third part of this work is devoted to the study of an inverse problem. The observations are discrete values
taken by different curves of unknown shape and we assume that they deduce the one from another by a translation parameter. The objective is to estimate the law of this shift parameter. We provide nonparametric estimators of the law at stake for different regularities and a practical simulation algorithm.