2021/03/12 by Claire Lacour, Lacour, Claire, Thanh Mai Pham Ngoc +1
Mathematics · #FOS: Mathematics #Statistics Theory (math.ST) #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.2103.07318
Electronic Journal of Statistics , Shaker Heights, OH : Institute of Mathematical Statistics, In press
arxiv created 2022/05/31 · arxiv updated 2022/06/01
We consider X 1 ,. .. , X n a sample of data on the circle S 1 , whose distribution 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 ∈ (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.