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Abstract

We study a method for estimating a density f in the d-dimensional Euclidean space 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 the density contour clusters. We assume that the density contourclusters lie in a given class of measurable subsets of the d-dimensionalEuclidean space. By choosing this class appropriately it is possible tomodel for example monotonicity, symmetry or multimodality. The mainmathematical tool for proving consistency and rates of convergence of thedensity estimator is empirical process theory.