Parameter recovery in two-component contamination mixtures: the $\mathbb{L}^2$ strategy

TL;DR

This paper investigates the estimation of parameters in a two-component contamination mixture model with known shape, establishing optimal convergence rates and highlighting limitations of classical parametric rates when parameters tend to zero.

Contribution

It provides the first analysis of optimal convergence rates for mixture parameters in a contamination model with known shape, revealing when classical rates are unattainable.

Findings

Optimal convergence rates for mixture parameter estimation are established.

Classical parametric rate $1/\sqrt{n}$ is not achievable when parameters tend to zero.

The shape of the contamination distribution is assumed known throughout.

Abstract

In this paper, we consider a parametric density contamination model. We work with a sample of i.i.d. data with a common density, $f^\star =(1-\lambda^\star) \phi + \lambda^\star \phi(.-\mu^\star)$, where the shape $\phi$ is assumed to be known. We establish the optimal rates of convergence for the estimation of the mixture parameters $(\lambda^\star,\mu^\star)$. In particular, we prove that the classical parametric rate $1/\sqrt{n}$ cannot be reached when at least one of these parameters is allowed to tend to $0$ with $n$.