Semiparametric inference for mixtures of circular data

Abstract

We consider $X_1,\dots ,X_n$ a sample of data on the circle ${\mathbb{S}}^1$, whose distribution is a two-component mixture. Denoting R and Q two rotations on ${\mathbb{S}}^1$, 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 this paper we estimate both the parametric part $\mathit{\theta }=(p,R,Q)$ 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.