Exact adaptive estimation of the shape of a periodic function with unknown period corrupted by white noise
Ismael Castillo, Celine Levy-Leduc and Catherine Matias
Mathematical Methods of Statistics 2005.

## Abstract

We consider the nonparametric estimation of the shape of a periodic function with unknown period $\theta$, observed in the presence of additive Gaussian white noise. In this semiparametric framework, estimators of the period with a parametric rate of convergence have been proposed in \cite{Golubev88}, \cite{Castillo05} and \cite{GassiatLevyLeduc06}. The existence of such a preliminary estimator of $\theta$ enables us to introduce an estimation procedure of the shape of the periodic function, using Stein's blockwise method. This estimator is sharp minimax adaptive on a scale of a family of Sobolev classes. The results are illustrated on a simulation study and compared with blind methods which do not use the periodicity assumption on the signal.