We consider a sample of data on the circle , whose distribution is a two-component mixture. Denoting R and Q two rotations on , the density of the ’s is assumed to be , where and f is an unknown density on the circle. In this paper we estimate both the parametric part and the nonparametric part f. The specific problems of identifiability on the circle are studied. A consistent estimator of θ is introduced and its asymptotic normality is proved. We propose a Fourier-based estimator of f with a penalized criterion to choose the resolution level. We show that our adaptive estimator is optimal from the oracle and minimax points of view when the density belongs to a Sobolev ball. Our method is illustrated by numerical simulations.
The authors would like to thank the Editors and one anonymous referee for valuable comments and suggestions leading to corrections and improvements of the article.
"Semiparametric inference for mixtures of circular data." Electron. J. Statist. 16 (1) 3482 - 3522, 2022. https://doi.org/10.1214/22-EJS2024