Density Estimation under Qualitative Assumptions in Higher Dimensions

We study a method for estimating a density f in Rd under assumptions which are of qualitative nature. The resulting density estimator can be considered as a generalization of the Grenander estimator for monotone densities. The assumptions on f are given in terms of shape restrictions of the density contour clusters Γ(λ) = (x : f(x) ≥ λ). We assume that for all λ ≥ 0 the sets Γ(λ) lie in a given class C of measurable subsets of Rd. By choosing C appropriately it is possible to model for example monotonicity, symmetry, or multimodality. The main mathematical tool for proving consistency and rates of convergence of the density estimator is empirical process theory. It turns out that the rates depend on the richness of C measured by metric entropy.