Minimax optimal estimation in partially linear additive models under high dimension
Minimax optimal estimation in partially linear additive models under high dimension
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DOI：
10.3150/18-bej1021
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发表时间：
2016-12
期刊：
影响因子：
1.5
通讯作者：
Zhuqing Yu;M. Levine;Guang Cheng
中科院分区：
文献类型：
--
作者：
Zhuqing Yu;M. Levine;Guang Cheng
本文导出了具有高维稀疏向量和光滑泛函分量的部分线性加性模型的参数和非参数分量估计的极小极大率。欧几里得分量的极大极小下界是典型的稀疏估计率，它与非参数光滑性指标无关。然而，每个分量函数的极大极小下界表现出参数分量的维数和稀疏度与相关非参数分量的平滑度之间的相互作用。事实上，当非参数分量的平滑度或参数分量的维数足够大时，平滑非参数估计的最小最大风险可以降低到稀疏估计率。在上面的设置中，我们证明了惩罚最小二乘估计几乎可以实现极大极小下界。
In this paper, we derive minimax rates for estimating both parametric and nonparametric components in partially linear additive models with high dimensional sparse vectors and smooth functional components. The minimax lower bound for Euclidean components is the typical sparse estimation rate that is independent of nonparametric smoothness indices. However, the minimax lower bound for each component function exhibits an interplay between the dimensionality and sparsity of the parametric component and the smoothness of the relevant nonparametric component. Indeed, the minimax risk for smooth nonparametric estimation can be slowed down to the sparse estimation rate whenever the smoothness of the nonparametric component or dimensionality of the parametric component is suffciently large. In the above setting, we demonstrate that penalized least square estimators can nearly achieve minimax lower bounds.