Parameter recovery in two-component contamination mixtures: the L2 strategy

In this paper, we consider a parametric density contamination model. We work with a sample of i.i.d. data with a common density, $f^{\star }=(1-\lambda^{\star })\phi +\lambda^{\star }\phi (\cdot-\mu^{\star })$, where the shape $\phi $ is assumed to be known. We establish the optimal rates of convergence for the estimation of the mixture parameters $(\lambda^{\star },\mu^{\star })\in (0,1)\times \mathbb{R}^{d}$. 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 $n$.