Optimal third root asymptotic bounds in the statistical estimation of thresholds

• 成果类型: Article
• 署名作者: Merkl, Franz; Mohammadi, Leila
• 署名单位: University of Munich; Leiden University; Leiden University Medical Center (LUMC); Leiden University - Excl LUMC
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053607000000325
• 发表日期: 2007
• 页码: 2193-2218
• 关键词: Nonparametric regression; CONVERGENCE; rates

摘要

This paper is concerned with estimating the intersection point of two densities, given a sample of both of the densities. This problem arises in classification theory. The main results provide lower bounds for the probability of the estimation errors to be large on a scale determined by the inverse cube root of the sample size. As corollaries, we obtain probabilistic bounds for the prediction error in a classification problem. The key to the proof is an entropy estimate. The lower bounds are based on bounds for general estimators, which are applicable in other contexts as well. Furthermore, we introduce a class of optimal estimators whose errors asymptotically meet the border permitted by the lower bounds.