光华讲坛——社会名流与企业家论坛第5480期

主题：Hypothesis Testing on Linear Structures of High Dimensional Covariance Matrix

主讲人：宾州州立大学 李润泽教授

主持人：周岭 副教授

时间：2019年7月5日（星期） 下午14:00-15:00

地点：西南财经大学柳林校区弘远楼408会议室

主办单位：统计研究中心 统计学院 科研处

主讲人简介：

李润泽是宾州州立大学讲席教授。他的研究方向包括高维数据建模，非参数回归，半参数回归及其统计学的应用。他是IMS,ASA和AAAS fellow。他曾担任Annals of Statistics的副主编、主编。现担任Journal of American Statistical Association的副主编。

主要内容：

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.

本文关注高维协方差结构的重要性测试，旨在开发一个统一的测试常用线性协方差结构的框架。我们首先构造线性协方差结构中涉及的参数的一致估计，然后基于用于协方差矩阵估计的熵损失和二次损失开发线性协方差结构的两个测试。为了研究所提出的测试的渐近性质，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，我们在无效和替代假设下推导出这两个测试的极限分布。我们进一步表明，基于二次损失的测试是渐近无偏的。我们进行蒙特卡罗模拟研究，以检验两个测试的有限样本性能。我们的仿真结果表明，极限零分布非常接近它们的零分布，相应的渐近临界值很好地保持了I类误差率。我们的数值比较表明，在控制I类错误率和功率方面，所提出的测试优于现有测试。我们的模拟表明，基于二次损失的测试似乎比基于熵损失的测试具有更好的功效。