ON ESTIMATION OF ISOTONIC PIECEWISE CONSTANT SIGNALS

• 成果类型: Article
• 署名作者: Gao, Chao; Han, Fang; Zhang, Cun-Hui
• 署名单位: University of Chicago; University of Washington; University of Washington Seattle; Rutgers University System; Rutgers University New Brunswick
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1792
• 发表日期: 2020
• 页码: 629-654
• 关键词: least-squares estimators; unimodal regression; risk bounds; rates; CONVERGENCE

摘要

Consider a sequence of real data points X-1, ..., X-n with underlying means theta(1)*, ..., theta(n)*. This paper starts from studying the setting that theta(i)* is both piecewise constant and monotone as a function of the index i. For this, we establish the exact minimax rate of estimating such monotone functions, and thus give a nontrivial answer to an open problem in the shape-constrained analysis literature. The minimax rate under the loss of the sum of squared errors involves an interesting iterated logarithmic dependence on the dimension, a phenomenon that is revealed through characterizing the interplay between the isotonic shape constraint and model selection complexity. We then develop a penalized least-squares procedure for estimating the vector theta* = (theta(1)*, ..., theta(n)*)(T). This estimator is shown to achieve the derived minimax rate adaptively. For the proposed estimator, we further allow the model to be misspecified and derive oracle inequalities with the optimal rates, and show there exists a computationally efficient algorithm to compute the exact solution.