Spatially adaptive density estimation by localised Haar projections - Florian Gach; Richard Nickl; Vladimir Spokoiny (Humboldt University of Berlin, Collaborative Research Center 649 - Economic Risk, 2011)
Given a random sample from some unknown density f0 : R → [0;∞) we devise Haar wavelet estimators for fo with variable resolution levels constructed from localised test procedures (as in Lepski, Mammen, and Spokoiny (1997, Ann. Statist.)). We show that these estimators adapt to spatially heterogeneous smoothness of f0, 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 practice is given.
Keywords: C14; 330; spatially inhomogeneous smoothness; bandwidth choice; propagation approach; Schätztheorie; Theorie