ON MINIMAX ESTIMATION OF A SPARSE NORMAL-MEAN VECTOR

• 成果类型: Article
• 署名作者: JOHNSTONE, IM
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176325368
• 发表日期: 1994
• 页码: 271-289
• 关键词: risk

摘要

Mallows has conjectured that among distributions which are Gaussian but for occasional contamination by additive noise, the one having least Fisher information has (two-sided) geometric contamination. A very similar problem arises in estimation of a nonnegative vector parameter in Gaussian white noise when it is known also that most [i.e., (1 - epsilon)) components are zero. We provide a partial asymptotic expansion of the minimax risk as epsilon --> 0. While the conjecture seems unlikely to be exactly true for finite epsilon, we verify it asymptotically up to the accuracy of the expansion. Numerical work suggests the expansion is accurate for epsilon as large as 0.05. The best l1-estimation rule is first- but not second-order minimax. The results bear on an earlier study of maximum entropy estimation and various questions in robustness and function estimation using wavelet bases.