Lp adaptive density estimation

Abstract

We provide global adaptive wavelet-type density estimates. Our procedures illustrate the refinement which can be obtained by replacing the Fourier basis by the wavelet basis in estimation methods. The main argument consists in observing that the estimated total energy of the details of a specified level j will be smaller or greater than some known threshold if precisely j is above or below the theoretical optimal level calculated with the a priori knowledge of the regularity of the density. This balancing effect leads directly to an adaptation procedure, and some natural extensions. We investigate the minimax properties of these procedures and explain their evolution for different global error measures.