Efficient Semiparametric Estimation of the Periods in a Superposition of Periodic Functions with Unknown Shape

We consider the estimation of the periods of periodic functions when their shape is unknown and they are corrupted by Gaussian white noise. In the case of a single periodic function, we propose a consistent and asymptotically efficient semiparametric estimator of the period. We then study the case of a sum of two periodic functions of unknown shape with different periods and propose semiparametric estimators of their periods that are consistent and asymptotically Gaussian.