Consider the problem of estimating the $\gamma$-level set$G^*_{\gamma}=\{x:f(x)\geq\gamma\}$ of an unknown $d$-dimensional densityfunction $f$ based on $n$ independent observations $X_1,...,X_n$ from thedensity. This problem has been addressed under global error criteria related tothe symmetric set difference. However, in certain applications a spatiallyuniform mode of convergence is desirable to ensure that the estimated set isclose to the target set everywhere. The Hausdorff error criterion provides thisdegree of uniformity and, hence, is more appropriate in such situations. It isknown that the minimax optimal rate of error convergence for the Hausdorffmetric is $(n/\log n)^{-1/(d+2\alpha)}$ for level sets with boundaries thathave a Lipschitz functional form, where the parameter $\alpha$ characterizesthe regularity of the density around the level of interest. However, theestimators proposed in previous work are nonadaptive to the density regularityand require knowledge of the parameter $\alpha$. Furthermore, previouslydeveloped estimators achieve the minimax optimal rate for rather restrictedclasses of sets (e.g., the boundary fragment and star-shaped sets) thateffectively 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 fullydata-driven procedure that is adaptive to unknown regularity conditions andachieves near minimax optimal Hausdorff error control for a class of densitylevel sets with very general shapes and multiple connected components.
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