Spatially adaptive density estimation by localised Haar projections

Given a random sample from some unknown density f(0):R -> [0, infinity) we devise Haar wavelet estimators for f(0) with variable resolution levels constructed from localised test procedures (as in Lepski, Mammen and Spokoiny (Ann. Statist. 25 (1997) 927-947)). We show that these estimators satisfy an oracle inequality that adapts to heterogeneous smoothness of f(0), simultaneously for every point x in a fixed interval, in sup-norm loss. The thresholding constants involved in the test procedures can be chosen in practice under the idealised assumption that the true density is locally constant in a neighborhood of the point x of estimation, and an information theoretic justification of this practise is given.