Semiparametric inference for mixtures of circular data

Claire Lacour (1) , Thanh Mai Pham Ngoc (2)

1 LAMA - Laboratoire Analyse et de Mathématiques Appliquées
2 LMO - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: Circular data; Semiparametric estimation; Mixture model

Domaines: Statistiques [math.ST]

Claire Lacour, Thanh Mai Pham Ngoc. Semiparametric inference for mixtures of circular data. Electronic Journal of Statistics , 2022, 16 (1), pp.3482-3522. ⟨hal-03149434v2⟩