Adjusted location‐invariant U‐tests for the covariance matrix with elliptically high‐dimensional data

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 p$$ p $$ may go to infinity in an arbitrary way as the sample size n$$ n $$ 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 (n,p)$$ \left(n,p\right) $$ 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.