Spatially Adaptive Density Estimation by Localised Haar Projections

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Given a random sample from some unknown density f₀: R → [0, ∞) we devise Haar wavelet estimators for f₀ 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.