An adaptive moment estimator of a parameter of a distribution constructed from observations with admixture

We consider the problem of estimating an unknown parameter from observations with an admixture. The concentration of the admixture is varying with observations and assumed to be known, while its distribution is unknown. We study moment estimators and prove that they are consistent and asymptotically normal. We use an adaptive technique that allows us to determine estimators whose asymptotic variance is minimal among moment estimators.