OPTIMAL ESTIMATION OF HIGH-DIMENSIONAL GAUSSIAN LOCATION MIXTURES

• 成果类型: Article
• 署名作者: Doss, Natalie; Wu, Yihong; Yang, Pengkun; Zhou, Harrison H.
• 署名单位: Yale University; Tsinghua University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/22-AOS2207
• 发表日期: 2023
• 页码: 62-95
• 关键词: CONVERGENCE-RATES; maximum-likelihood; strong identifiability; parameter-estimation; dirichlet mixtures; Finite; ORDER; decompositions; ratio; rank

摘要

This paper studies the optimal rate of estimation in a finite Gaussian location mixture model in high dimensions without separation conditions. We assume that the number of components k is bounded and that the centers lie in a ball of bounded radius, while allowing the dimension d to be as large as the sample size n. Extending the one-dimensional result of Heinrich and Kahn (Ann. Statist. 46 (2018) 2844-2870), we show that the minimax rate of estimating the mixing distribution in Wasserstein distance is Theta((d/n)(1/4) + n(-1)/((4k-2))), achieved by an estimator computable in time O(nd(2) + n(5/4)). Furthermore, we show that the mixture density can be estimated at the optimal parametric rate Theta(root d/n) in Hellinger distance and provide a computationally efficient algorithm to achieve this rate in the special case of k = 2.