摘要：We consider X1,…,Xn a sample of data on the circle S1, whose distribution is a two-component mixture. Denoting R and Q two rotations on S1, the density of the Xi’s is assumed to be g(x)=pf(R−1x)+(1−p)f(Q−1x), where p∈(0,1) and f is an unknown density on the circle. In this paper we estimate both the parametric part θ=(p,R,Q) 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.
