Hypothesis Testing Of The Proportionality Of Two Covariance Matrices

In multivariate statistical analysis,the hypothesis testings of the covariance matrix include the sphericity test,the identity test and the test of the equality of covariance matrices.We can also test the proportionality of covariance matrices,which can be applied in many fields,for example,the quantitative genetic experi-ment called paternal half-sib design,discriminant analysis,and principal components analysis.In high-dimensional hypothesis testings,most of classic methods are not valid so that we need to find new statistics.In chapter 1 and 2,we introduce some methods for testing of high-dimensional covariance matrix.Both unbiased estimators and their asymptotical distributions of the trace of covariance matrices are given in chapter 3.In chapter 4,based on statistics for testing the equality of covariance matrices proposed by Schott and Sri-vastava,we propose statistics T1,T2 and T3 to test the proportionality of covariance matrices and derive their asymptotical distributions.According to Cauchy-Schwarz inequality,we propose new statistic T4,which converges to a normal distribution.In chapter 5,we further study properties of four statistics by simulations.First-ly,in the case c is known,the empirical sizes of T1 and T2 are close to 0.05 and the empirical powers increase to 1 as(p,n1,n2)increase.But in most cases,the empiri-cal powers of T1 are better than those of T2.Secondly,in the case c is unknown,the empirical sizes of T3 and T4 axe close to 0.05 and the empirical powers increase to 1 as(p,n1,n2)increase.Specially,T4 is more valid than T3 under spiked structures.Thirdly,we compare T3,T4 with TLBS proposed by Liu et al..Simulation results show that the empirical powers of T3 and T4 are equal to 1,while the empirical powers of TLBS is close to 1 slowly as(p,n1,n2)increase.