Density estimation of a mixture distribution with unknown point-mass and normal error

We consider the model Y=X+ξ where Y is observable, ξ is a noise random variable with density fξ, X has an unknown mixed density such that P(X=Xc)=1−p, P(X=a)=p with Xc being continuous and p∈(0,1), a∈R. Typically, in the last decade, the model has been widely considered in a number of papers for the case of fully known quantities a,fξ. In this paper, we relax the assumptions and consider the parametric error ξ∼σN(0,1) with an unknown σ>0. From i.i.d. copies Y1,…,Ym of Y we will estimate (σ,p,a,fXc) where fXc is the density of Xc. We also find the lower bound of convergence rate and verify the minimax property of established estimators.