Semiparametric inference for mixtures of circular data

We consider X-1, ..., X-n a sample of data on the circle S-1, whose distribution is a two-component mixture. Denoting R and Q two rotations on S-1, the density of the X-i's is assumed to be g(x) = pf (R(-1)x)+(1 - p)f(Q(-1)x), where p is an element of (0, 1) and f is an unknown density on the circle. In this paper we estimate both the parametric part theta = (p, R, Q) and the nonparametric part f. The specific problems of identifiability on the circle are studied. A consistent estimator of theta 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.