Testing Linear Structure Of High-dimensional Covariance Matrix Based On Difference Of Matrices

High dimensional covariance matrix plays a important role in finance,economics,biology and other fileds.In multivariate analysis,both the sphericity testing and testing whether the covariance matrix equals to a known one are given considerable attention.In this paper,a new statistic is proposed to test the linear structure of the covariance matrix based on the existing literature.We first construct a consistent estimation for parameters involved in the linear structure and then develop a new test T_nfor the linear covariance structures based on the quadratic loss.Applying some results of the large dimensional random matrix theory,we derive the limiting distribution of the test we proposed and conduct Monte Carlo simulation study to examine the finite sample per-formance of the test.The results show that the asymptotic distribution approximates the null distribution quite well and corresponding asymptotic critical values keep type error rate very well.Our numerical comparison implies that our proposed test seems to have better power than the existing tests when there are great difference between the null hypothesis and the alternative hypothesis.