Adaptive estimation of convex and polytopal density support

• 成果类型: Article
• 署名作者: Brunel, Victor-Emmanuel
• 署名单位: Institut Polytechnique de Paris; ENSAE Paris; University of Haifa
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0605-5
• 发表日期: 2016
• 页码: 1-16
• 关键词: boundary; rates; sets; hull

摘要

We estimate the support of a uniform density, when it is assumed to be a convex polytope or, more generally, a convex body in R-d. In the polytopal case, we construct an estimator achieving a rate which does not depend on the dimension d, unlike the other estimators that have been proposed so far. For d >= 3, our estimator has a better risk than the previous ones, and it is nearly minimax, up to a logarithmic factor. We also propose an estimator which is adaptive with respect to the structure of the boundary of the unknown support.