Adaptive Hausdorff Estimation of Density Level Sets

Hausdorff accurate estimation of density level sets is relevant in applications where a spatially uniform mode of convergence is desired to ensure that the estimated set is close to the target set at all points. The minimax optimal rate of error convergence for the Hausdorff metric is known to be (n/ logn) for level sets with Lipschitz boundaries, where the parameter α characterizes the regularity of the density around the level of interest. However, all previous work assumes knowledge of the density regularity as characterized by the parameter α. Moreover, the estimators proposed in previous work achieve the minimax optimal rate for rather restricted classes of sets (for example, the boundary fragment and star-shaped sets) that effectively reduce the set estimation problem to a function estimation problem. This characterization precludes level sets with multiple connected components, which are fundamental to many applications. This paper presents a fully data-driven procedure that is adaptive to unknown local density regularity, and achieves minimax optimal Hausdorff error control for a class of level sets with very general shapes and multiple connected components.