Sharp oracle inequalities for Least Squares estimators in shape restricted regression
Sharp oracle inequalities for Least Squares estimators in shape restricted regression
复制标题
形状限制回归中最小二乘估计量的尖锐预言不等式
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
P. Bellec
中科院分区：
文献类型：
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作者：
P. Bellec
最小二乘（LS）估计量的性能在等渗，单峰和凸回归中进行了研究。我们的结果具有尖锐的甲骨文不平等现象，这些不平等现象是模型错误指定错误。在等渗和单峰回归中，LS估计器实现了非参数率$ n^{ - 2/3} $，以及订单$ k/n $的参数率，而对数因素，其中$ k $是常数的数量真实参数的部分。
在单变量凸回归中，LS估计器满足了对数$ q/n $的自适应风险限制，其中$ q $是真实回归函数的仿射件的数量。这种自适应风险绑定均适用于任何设计点。虽然Guntuboyina和Sen（2013）确定凸回归的非参数率是$ n^{ - 4/5} $的均值，但我们表明，凸回归的非参数速率可以慢于$ n^ {-2/3} $对于某些最差的设计点。这种现象可以解释如下：尽管凸性带来的结构比单偶像性更多，但对于某些最差的设计点，这种额外的结构是无信息的，单峰回归和凸回归的非参数速率都是$ n^{ - 2/3} $ 。
The performance of Least Squares (LS) estimators is studied in isotonic, unimodal and convex regression. Our results have the form of sharp oracle inequalities that account for the model misspecification error. In isotonic and unimodal regression, the LS estimator achieves the nonparametric rate $n^{-2/3}$ as well as a parametric rate of order $k/n$ up to logarithmic factors, where $k$ is the number of constant pieces of the true parameter.
In univariate convex regression, the LS estimator satisfies an adaptive risk bound of order $q/n$ up to logarithmic factors, where $q$ is the number of affine pieces of the true regression function. This adaptive risk bound holds for any design points. While Guntuboyina and Sen (2013) established that the nonparametric rate of convex regression is of order $n^{-4/5}$ for equispaced design points, we show that the nonparametric rate of convex regression can be as slow as $n^{-2/3}$ for some worst-case design points. This phenomenon can be explained as follows: Although convexity brings more structure than unimodality, for some worst-case design points this extra structure is uninformative and the nonparametric rates of unimodal regression and convex regression are both $n^{-2/3}$.
影响因子：
1.6
作者：
Amelunxen, Dennis;Lotz, Martin;Tropp, Joel A.
通讯作者：
Tropp, Joel A.