OPTIMAL ADAPTIVE ESTIMATION OF LINEAR FUNCTIONALS UNDER SPARSITY

• 成果类型: Article
• 署名作者: Collier, Olivier; Comminges, Laetitia; Tsybakov, Alexandre B.; Verzelen, Nicolas
• 署名单位: Universite Paris Saclay; Universite PSL; Universite Paris-Dauphine; Institut Polytechnique de Paris; ENSAE Paris; Ecole Polytechnique; INRAE
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1653
• 发表日期: 2018
• 页码: 3130-3150
• 关键词: minimax estimation

摘要

We consider the problem of estimation of a linear functional in the Gaussian sequence model where the unknown vector theta is an element of R-d belongs to a class of s-sparse vectors with unknown s. We suggest an adaptive estimator achieving a nonasymptotic rate of convergence that differs from the minimax rate at most by a logarithmic factor. We also show that this optimal adaptive rate cannot be improved when s is unknown. Furthermore, we address the issue of simultaneous adaptation to s and to the variance sigma(2) of the noise. We suggest an estimator that achieves the optimal adaptive rate when both s and sigma(2) are unknown.