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We consider X 1 ,. .. , X n a sample of data on the circle S 1 , whosedistribution is a twocomponent 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 $\in$ (0, 1) and f is an unknown density on the circle. In thispaper we estimate both the parametric part $\theta$ = (p, R, Q) and thenonparametric part f. The specific problems of identifiability on the circleare studied. A consistent estimator of $\theta$ is introduced and itsasymptotic normality is proved. We propose a Fourier-based estimator of f witha penalized criterion to choose the resolution level. We show that our adaptiveestimator is optimal from the oracle and minimax points of view when thedensity belongs to a Sobolev ball. Our method is illustrated by numericalsimulations.

Semiparametric inference for mixtures of circular data