Adjusted location‐invariant U‐tests for the covariance matrix with elliptically high‐dimensional data
研究了椭圆分布下协方差矩阵的几种U检验，发现经典检验在一般椭圆总体中表现不佳，提出修正方法并给出渐近零分布，通过模拟和实证验证了准确性。
Abstract This paper analyzes several covariance matrix U‐tests, which are constructed by modifying the classical John‐Nagao and Ledoit‐Wolf tests, under the elliptically distributed data structure. We study the limiting distributions of these location‐invariant test statistics as the data dimension may go to infinity in an arbitrary way as the sample size does. We find that they tend to have unsatisfactory size performances for general elliptical population. This is mainly because such population often possesses high‐order correlations among their coordinates. Taking such kind of dependency into consideration, we propose necessary corrections for these tests to cope with elliptically high‐dimensional data. For computational efficiency, alternative forms of the new test statistics are also provided. We derive the universal asymptotic null distributions of the proposed test statistics under elliptical distributions and beyond. The powers of the proposed tests are further investigated. The accuracy of the tests is demonstrated by simulations and an empirical study.