密度水平集的自适应Hausdorff估计
Adaptive Hausdorff estimation of density level sets
摘要：
考虑基于来自密度的$n$独立观测$X_1,...,X_n$估计未知$d$维密度函数$f$的$γ$水平集$G^*_γ=\{x:f(x)\geqγ\}$的问题.这个问题已经解决了下的全局误差标准有关的对称集差。然而，在某些应用中，期望收敛的空间均匀模式以确保估计集在任何地方都接近目标集。豪斯多夫误差准则提供了这种程度的一致性，因此，在这种情况下更合适。已知对于具有Lipschitz函数形式的边界的水平集，Hausdorff度量的极小极大最优误差收敛率是$(n/\log n)^{-1/(d+2α)}$，其中参数$α$表征感兴趣水平周围的密度的规律性。然而，在以前的工作中提出的估计是不适应的密度规律性，并需要知识的参数$α$。此外，以前开发的估计器实现了极小极大最优速率，而限制类集（例如，边界片段和星形集合），其有效地将集合估计问题简化为函数估计问题。这种特征排除了具有多个连通分量的水平集，这是许多应用的基础。本文提出了一个完全数据驱动的过程，是自适应未知的正则性条件，并实现了近极小极大最优Hausdorff差错控制的一类密度水平集具有非常一般的形状和多个连通组件。
英文摘要：
Consider the problem of estimating the $γ$-level set $G^*_γ=\{x:f(x)\geqγ\}$ of an unknown $d$-dimensional density function $f$ based on $n$ independent observations $X_1,...,X_n$ from the density. This problem has been addressed under global error criteria related to the symmetric set difference. However, in certain applications a spatially uniform mode of convergence is desirable to ensure that the estimated set is close to the target set everywhere. The Hausdorff error criterion provides this degree of uniformity and, hence, is more appropriate in such situations. It is known that the minimax optimal rate of error convergence for the Hausdorff metric is $(n/\log n)^{-1/(d+2α)}$ for level sets with boundaries that have a Lipschitz functional form, where the parameter $α$ characterizes the regularity of the density around the level of interest. However, the estimators proposed in previous work are nonadaptive to the density regularity and require knowledge of the parameter $α$. Furthermore, previously developed estimators achieve the minimax optimal rate for rather restricted classes of sets (e.g., 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 regularity conditions and achieves near minimax optimal Hausdorff error control for a class of density level sets with very general shapes and multiple connected components.
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