A unified treatment of direct and indirect estimation of a probability density and its derivatives

This paper presents convolution-based estimates of a probability density and its derivatives. The proposed estimates can handle either contaminated data or not and they comprehend some classical estimates such that kernel, regularization estimates. By putting these direct and indirect estimation problems in the same framework, we clearly see how the estimates performances are affected by contamination and by the order of the derivative to be estimated. Minimax optimal rates for the MISE criterion are proposed.