Corrections To The Score Tests On Large Dimensional Sample Covariance Matrices Structure

With the advent of the era of big data, more and more massive high-dimensional data appeared in various scientific fields, which needs statisticians to provide some appropriate predictions and instructive recommendations based on the data. How-ever, since the classical theories of multivariate statistical analysis are based on the assumption that the variable dimension p is much less than the sample size n, so many of the classic limiting tools can only be applied to the small dimension. Thus, it bring a serious lack of the adaptive theoretical tools for high-dimensional data statistical analysis in practice. Therefore, this problem attracts much more attentions of the statisticians recently. Over the last decade, there have been a lot of statistical methods for high-dimensional data emerged, which compensates the serious effects due to the increase in dimension. In this paper, we also mentioned a number of classical multivariate statistical theories, which play a poor performance of high-dimensional data, or even fail completely, including Rao (1948) score test and Wald (1943) score test. The article first introduces the classical Rao score test and Wald score test, and derives two score test statistics for the variance test. Based on it, the corrected score tests for the hypotheses of covariance matrices are proposed by the application of large dimensional random matrix theory. Then the simulations are conducted to compare our new score tests and other large di-mensional test methods for the test of covariance matrix, which shows that the corrected score tests can be more widely used in large-dimensional non-Gaussian data and arrive at a good test result.