Semiparametric inference for mixtures of circular data

TL;DR

This paper develops a method to estimate both the mixing parameters and the unknown density in a mixture of circular data, demonstrating its theoretical optimality and practical effectiveness through simulations.

Contribution

It introduces a novel Fourier-based adaptive estimator for the mixture model on the circle, handling identifiability and achieving minimax optimality.

Findings

Estimator of mixing parameters is consistent and asymptotically normal.

Adaptive Fourier estimator achieves minimax optimal rates.

Numerical simulations confirm the method's effectiveness.

Abstract

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 $\in$ (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.