On estimation of Lr-norms in Gaussian white noise models

• 成果类型: Article
• 署名作者: Han, Yanjun; Jiao, Jiantao; Mukherjee, Rajarshi
• 署名单位: Stanford University; University of California System; University of California Berkeley; Harvard University; Harvard T.H. Chan School of Public Health
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-00982-x
• 发表日期: 2020
• 页码: 1243-1294
• 关键词: minimax estimation; integral functionals; Adaptive estimation; density; hypotheses

摘要

We provide a complete picture of asymptotically minimax estimation of L-r-norms (for any r >= 1) of the mean in Gaussian white noise model over Nikolskii-Besov spaces. In this regard, we complement the work of Lepski et al. (Probab Theory Relat Fields 113(2):221-253, 1999), who considered the cases of r = 1 (with poly-logarithmic gap between upper and lower bounds) and r even (with asymptotically sharp upper and lower bounds) over Holder spaces. We additionally consider the case of asymptotically adaptive minimax estimation and demonstrate a difference between even and non-even r in terms of an investigator's ability to produce asymptotically adaptive minimax estimators without paying a penalty.