Direct density estimation as an ill-posed inverse estimation problem

A possible definition of ill-posedness in statistical estimation is the lack of qualitative robustness. In this sense direct density estimation shares ill-posedness with the more obviously ill-posed indirect density estimation models, of which it is a special case. A general construction pattern for estimators is proposed, based on suitable preconditioning, that works for both direct and indirect density estimation. Special emphasis is on its application to the direct case, where in general it yields delta-sequence estimators. More specifically both kernel and series type estimators are included depending on the choice of preconditioning operator. In particular sinc and other flattop kernel estimators emerge in a natural way.