Nonparametric Shape-Restricted Regression
Nonparametric Shape-Restricted Regression
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DOI：
10.1214/18-sts665
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发表时间：
2018-11-01
影响因子：
5.7
通讯作者：
Sen, Bodhisattva
中科院分区：
文献类型：
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作者：
Guntuboyina, Adityanand;Sen, Bodhisattva
我们考虑形状约束下的非参数回归问题。主要的例子包括保序回归(关于任何偏序)、单峰/凸回归、加性形状约束回归和约束单指数模型。我们回顾了最小二乘估计(LSE)在这些问题中的一些理论性质，强调了LSE的自适应性质。特别是，我们研究了伦敦证券交易所的风险行为，以及它的逐点极限分布理论，特别强调了保序回归。我们综述了围绕这些形状限制函数构造逐点置信度区间的各种方法。我们还简要讨论了LSE的计算，并指出了一些有待解决的研究问题和未来的研究方向。
We consider the problem of nonparametric regression under shape constraints. The main examples include isotonic regression (with respect to any partial order), unimodal/convex regression, additive shape-restricted regression and constrained single index model. We review some of the theoretical properties of the least squares estimator (LSE) in these problems, emphasizing on the adaptive nature of the LSE. In particular, we study the behavior of the risk of the LSE, and its pointwise limiting distribution theory, with special emphasis to isotonic regression. We survey various methods for constructing pointwise confidence intervals around these shape-restricted functions. We also briefly discuss the computation of the LSE and indicate some open research problems and future directions.