HYPOTHESIS TESTING ON LINEAR STRUCTURES OF HIGH DIMENSIONAL COVARIANCE MATRIX.
HYPOTHESIS TESTING ON LINEAR STRUCTURES OF HIGH DIMENSIONAL COVARIANCE MATRIX.
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DOI：
10.1214/18-aos1779
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发表时间：
2019-12
影响因子：
4.5
通讯作者：
Shu-rong Zheng;Zhao Chen;H. Cui;Runze Li
中科院分区：
文献类型：
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作者：
Shu-rong Zheng;Zhao Chen;H. Cui;Runze Li
本文研究了高维协方差结构的显著性检验问题，旨在为检验常用的线性协方差结构提供一个统一的框架。首先构造了线性协方差结构中参数的一致性估计，然后分别基于熵损失和二次损失对线性协方差结构进行了检验.为了研究所提出的检验的渐近性质，我们研究了相关的高维随机矩阵理论，并建立了几个非常有用的渐近结果。借助于这些渐近结果，我们得到了这两个测试的极限分布下的零假设和备择假设。我们进一步证明了基于二次损失的检验是渐近无偏的。我们进行蒙特卡罗模拟研究，以检查有限样本的性能，这两个测试。模拟结果表明，极限零分布很好地逼近了零分布，并且相应的渐近临界值很好地保持了I型错误率。我们的数值比较表明，所提出的测试优于现有的控制I型错误率和功率。我们的模拟表明，基于二次损失的测试似乎有更好的权力比基于熵损失的测试。
This paper is concerned with test of significance on high dimensional covariance structures, and aims to develop a unified framework for testing commonly-used linear covariance structures. We first construct a consistent estimator for parameters involved in the linear covariance structure, and then develop two tests for the linear covariance structures based on entropy loss and quadratic loss used for covariance matrix estimation. To study the asymptotic properties of the proposed tests, we study related high dimensional random matrix theory, and establish several highly useful asymptotic results. With the aid of these asymptotic results, we derive the limiting distributions of these two tests under the null and alternative hypotheses. We further show that the quadratic loss based test is asymptotically unbiased. We conduct Monte Carlo simulation study to examine the finite sample performance of the two tests. Our simulation results show that the limiting null distributions approximate their null distributions quite well, and the corresponding asymptotic critical values keep Type I error rate very well. Our numerical comparison implies that the proposed tests outperform existing ones in terms of controlling Type I error rate and power. Our simulation indicates that the test based on quadratic loss seems to have better power than the test based on entropy loss.