Precision matrix, which is the inverse of covariance matrix, plays an important role in statistics, as it captures the partial correlation between variables. Testing the equality of two precision matrices in high dimensional setting is a very challenging but meaningful problem, especially in the differential network modelling. To our best knowledge, existing test is only powerful for sparse alternative patterns where two precision matrices differ in a small number of elements. In this paper we propose a data-adaptive test which is powerful against either dense or sparse alternatives. Multiplier bootstrap approach is utilized to approximate the limiting distribution of the test statistic. Theoretical properties including asymptotic size and power of the test are investigated. Simulation study verifies that the data-adaptive test performs well under various alternative scenarios. The practical usefulness of the test is illustrated by applying it to a gene expression data set associated with lung cancer.

中文翻译：

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高维二维样本精度矩阵测试：通过乘法器引导的自适应方法

精度矩阵是协方差矩阵的逆矩阵，它在统计中起着重要作用，因为它捕获了变量之间的偏相关。在高维设置中测试两个精度矩阵的相等性是一个非常具有挑战性但有意义的问题，尤其是在差分网络建模中。据我们所知，现有测试仅适用于稀疏替代模式，其中两个精度矩阵在少量元素上不同。在本文中，我们提出了一种数据自适应测试，它对密集或稀疏的替代方案都非常有效。乘数引导方法用于近似测试统计量的极限分布。研究了包括渐近大小和检验功效在内的理论特性。仿真研究验证了数据自适应测试在各种替代场景下的表现良好。通过将其应用于与肺癌相关的基因表达数据集来说明该测试的实际用途。