Optimal change-point estimation from indirect observations

• 成果类型: Article
• 署名作者: Goldenshluger, A.; Tsybakov, A.; Zeevi, A.
• 署名单位: University of Haifa; Columbia University; Sorbonne Universite; Universite Paris Cite
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053605000000750
• 发表日期: 2006
• 页码: 350-372
• 关键词: nonparametric deconvolution; Minimax Estimation; Inverse problems; Optimal Rates; regression; Wavelets; CONVERGENCE; density; noise; jump

摘要

We study nonparametric change-point estimation from indirect noisy observations. Focusing on the white noise convolution model, we consider two classes of functions that are smooth apart from the change-point. We establish lower bounds on the minimax risk in estimating the change-point and develop rate optimal estimation procedures. The results demonstrate that the best achievable rates of convergence are determined both by smoothness of the function away from the change-point and by the degree of ill-posedness of the convolution operator. Optimality is obtained by introducing a new technique that involves, as a key element, detection of zero crossings of an estimate of the properly smoothed second derivative of the underlying function.