On Degrees of Freedom of Projection Estimators With Applications to Multivariate Nonparametric Regression
On Degrees of Freedom of Projection Estimators With Applications to Multivariate Nonparametric Regression
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DOI：
10.1080/01621459.2018.1537917
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发表时间：
2019-04-23
影响因子：
3.7
通讯作者：
Sen, Bodhisattva
中科院分区：
文献类型：
--
作者：
Chen, Xi;Lin, Qihang;Sen, Bodhisattva
在本文中，我们考虑了具有多元预测因子的非参数回归问题。我们给出了未知回归函数的估计的自由度和发散性的特征，这些估计是线性约束二次优化过程的输出，即具有线性约束和/或二次惩罚的最小二乘准则的最小化。作为我们结果的特例，我们得到了许多非参数回归问题的自由度的显式表达式，例如有界保序回归、多元(惩罚)凸回归和加性全变差正则化。作为特例，我们的理论还给出了许多统计文献中研究得很好的估计的自由度的已知结果，如岭回归、Lasso和广义Lasso。通过最小化Stein的无偏风险估计，我们的结果可以很容易地用于选择估计过程中涉及的调整参数(S)。作为我们分析的副产品，我们得到了一般偏序集上的有界保序回归和保序回归之间的一个有趣的联系，这是独立感兴趣的。有关本文的详细信息，可在网上找到。
In this article, we consider the nonparametric regression problem with multivariate predictors. We provide a characterization of the degrees of freedom and divergence for estimators of the unknown regression function, which are obtained as outputs of linearly constrained quadratic optimization procedures; namely, minimizers of the least-squares criterion with linear constraints and/or quadratic penalties. As special cases of our results, we derive explicit expressions for the degrees of freedom in many nonparametric regression problems, for example, bounded isotonic regression, multivariate (penalized) convex regression, and additive total variation regularization. Our theory also yields, as special cases, known results on the degrees of freedom of many well-studied estimators in the statistics literature, such as ridge regression, Lasso and generalized Lasso. Our results can be readily used to choose the tuning parameter(s) involved in the estimation procedure by minimizing the Stein's unbiased risk estimate. As a by-product of our analysis we derive an interesting connection between bounded isotonic regression and isotonic regression on a general partially ordered set, which is of independent interest. for this article are available online.