Statistical inference in compound functional models

• 成果类型: Article
• 署名作者: Dalalyan, Arnak; Ingster, Yuri; Tsybakov, Alexandre B.
• 署名单位: Institut Polytechnique de Paris; ENSAE Paris; Saint Petersburg State Electrotechnical University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-013-0487-y
• 发表日期: 2014
• 页码: 513-532
• 关键词: Nonparametric regression; asymptotic equivalence; Sparse Estimation; Optimal Rates; white-noise; aggregation; selection

摘要

We consider a general nonparametric regression model called the compound model. It includes, as special cases, sparse additive regression and nonparametric (or linear) regression with many covariates but possibly a small number of relevant covariates. The compound model is characterized by three main parameters: the structure parameter describing the macroscopic form of the compound function, the microscopic sparsity parameter indicating the maximal number of relevant covariates in each component and the usual smoothness parameter corresponding to the complexity of the members of the compound. We find non-asymptotic minimax rate of convergence of estimators in such a model as a function of these three parameters. We also show that this rate can be attained in an adaptive way.