Linear Hypothesis Testing in Dense High-Dimensional Linear Models
Linear Hypothesis Testing in Dense High-Dimensional Linear Models
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DOI：
10.1080/01621459.2017.1356319
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发表时间：
2016-10
影响因子：
3.7
通讯作者：
Yinchu Zhu;Jelena Bradic
中科院分区：
文献类型：
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作者：
Yinchu Zhu;Jelena Bradic
摘要：我们提出了一种在高维线性模型中检验线性假设的方法。所提出的检验没有对模型的大小施加任何限制，即模型稀疏性或表示假设的加载向量。为测试高维回归参数的一般线性函数提供渐近有效的方法是极具挑战性的，特别是没有对非零元素的数量做出限制性或不可验证的假设。我们建议测试与新设计的重构回归相关的力矩条件，其中输入被转换和增强特征。这些新特征直接结合了零假设的结构。测试统计量是以这样一种方式构建的，即原始模型参数中缺乏稀疏性不会对我们的过程的理论证明提出问题。我们建立了对I型误差的渐近精确控制，而不对模型参数或表示线性假设的向量施加任何稀疏性假设。我们的方法在检测零假设偏差方面也显示出一定的最优性。我们通过一些数值和实际数据示例证明了所提出方法的良好有限样本性能。本文的补充材料可在网上获得。
ABSTRACT We propose a methodology for testing linear hypothesis in high-dimensional linear models. The proposed test does not impose any restriction on the size of the model, that is, model sparsity or the loading vector representing the hypothesis. Providing asymptotically valid methods for testing general linear functions of the regression parameters in high-dimensions is extremely challenging—especially without making restrictive or unverifiable assumptions on the number of nonzero elements. We propose to test the moment conditions related to the newly designed restructured regression, where the inputs are transformed and augmented features. These new features incorporate the structure of the null hypothesis directly. The test statistics are constructed in such a way that lack of sparsity in the original model parameter does not present a problem for the theoretical justification of our procedures. We establish asymptotically exact control on Type I error without imposing any sparsity assumptions on model parameter or the vector representing the linear hypothesis. Our method is also shown to achieve certain optimality in detecting deviations from the null hypothesis. We demonstrate the favorable finite-sample performance of the proposed methods, via a number of numerical and a real data example. Supplementary materials for this article are available online.