Adaptive Dantzig density estimation

The aim of this paper is to build an estimate of an unknown density as a linear combination of functions of a dictionary. Inspired by Candès and Tao's approach, we propose a minimization of the l₁-norm of the coefficients in the linear combination under an adaptive Dantzig constraint coming from sharp concentration inequalities. This allows to consider a wide class of dictionaries. Under local or global structure assumptions, oracle inequalities are derived. These theoretical results are transposed to the adaptive Lasso estimate naturally associated to our Dantzig procedure. Then, the issue of calibrating these procedures is studied from both theoretical and practical points of view. Finally, a numerical study shows the significant improvement obtained by our procedures when compared with other classical procedures.