Sharp adaptive estimation of quadratic functionals

• 成果类型: Article
• 署名作者: Klemela, J
• 署名单位: University of Mannheim
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-005-0447-2
• 发表日期: 2006
• 页码: 539-564
• 关键词: nonparametric-estimation; asymptotic equivalence; pointwise; regression

摘要

Estimation of a quadratic functional of a function observed in the Gaussian white noise model is considered. A data-dependent method for choosing the amount of smoothing is given. The method is based on comparing certain quadratic estimators with each other. It is shown that the method is asymptotically sharp or nearly sharp adaptive simultaneously for the regular and irregular region. We consider l(p) bodies and construct bounds for the risk of the estimator which show that for p=4 the estimator is exactly optimal and for example when p is an element of [3,100], then the upper bound is at most 1.055 times larger than the lower bound. We show the connection of the estimator to the theory of optimal recovery. The estimator is a calibration of an estimator which is nearly minimax optimal among quadratic estimators.