Nonquadratic estimators of a quadratic functional

• 成果类型: Article
• 署名作者: Cai, TT; Low, MG
• 署名单位: University of Pennsylvania
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053605000000147
• 发表日期: 2005
• 页码: 2930-2956
• 关键词: squared density derivatives; Adaptive estimation; Minimax Estimation; confidence sets; regression; RISK

摘要

Estimation of a quadratic functional over parameter spaces that are not quadratically convex is considered. It is shown, in contrast to the theory for quadratically convex parameter spaces, that optimal quadratic rules are often rate suboptimal. In such cases minimax rate optimal procedures are constructed based on local thresholding. These nonquadratic procedures are sometimes fully efficient even when optimal quadratic rules have slow rates of convergence. Moreover, it is shown that when estimating a quadratic functional nonquadratic procedures may exhibit different elbow phenomena than quadratic procedures.