GENERALIZED CLUSTER TREES AND SINGULAR MEASURES

• 成果类型: Article
• 署名作者: Chen, Yen-Chi
• 署名单位: University of Washington; University of Washington Seattle
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1744
• 发表日期: 2019
• 页码: 2174-2203
• 关键词: density; Consistency; STABILITY; HOMOLOGY; uniform; sets

摘要

In this paper we study the alpha-cluster tree (alpha-tree) under both singular and nonsingular measures. The alpha-tree uses probability contents within a set created by the ordering of points to construct a cluster tree so that it is well defined even for singular measures. We first derive the convergence rate for a density level set around critical points, which leads to the convergence rate for estimating an alpha-tree under nonsingular measures. For singular measures, we study how the kernel density estimator (KDE) behaves and prove that the KDE is not uniformly consistent but pointwise consistent after rescaling. We further prove that the estimated alpha-tree fails to converge in the L-infinity metric but is still consistent under the integrated distance. We also observe a new type of critical points-the dimensional critical points (DCPs)-of a singular measure. DCPs are points that contribute to cluster tree topology but cannot be defined using density gradient. Building on the analysis of the KDE and DCPs, we prove the topological consistency of an estimated alpha-tree.