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

We consider the model where is observable, is a noise random variable with density , has an unknown mixed density such that , with being continuous and , . Typically, in the last decade, the model has been widely considered in a number of papers for the case of fully known quantities . In this paper, we relax the assumptions and consider the parametric error with an unknown . From i.i.d. copies of we will estimate where is the density of . We also find the lower bound of convergence rate and verify the minimax property of established estimators.