In this paper, we observe a fixed number of unknown 2π-periodic functions differing from each other by both phases and amplitude. This semiparametric model appears in literature under the name “shape invariant model.” While the common shape is unknown, we introduce an asymptotically efficient estimator of the finite-dimensional parameter (phases and amplitude) using the profile likelihood and the Fourier basis. Moreover, this estimation method leads to a consistent and asymptotically linear estimator for the common shape.
"Efficient estimation for a subclass of shape invariant models." Ann. Statist. 38 (3) 1885 - 1912, June 2010. https://doi.org/10.1214/07-AOS566