| With the rapid development of computer technology and modern information technology,high-dimensional data analysis has been developed and widely used,especially in the fields of financial stocks,genomics,environmental science,and image processing.However,the processing of such high-dimensional data poses a huge challenge to classical statistics.Traditional multivariate statistical analysis is mostly based on large sample theories,which has certain limitations for high-dimensional situations,and either cannot be used or has a low potential.Therefore,we need a new method to deal with these "big p,small n" data.For the test of two-sample covariance matrix,the classical likelihood ratio test is usually effective and powerful under low-dimensional and normal distribution.However,when the dimensionality is greater than the sample size,it cannot be applied in theory and practice.This thesis proposes two high-dimensional covariance matrix test methods based on permutation,and conducts theoretical research and simulation analysis on the test of high-dimensional two-sample covariance matrix.Through numerical simulation,the effect of the test method is verified.The main research contents are as follows:(1)Permutation test is a robust non-parametric hypothesis test method.It is better than ordinary parameter test when the sample size is small.Its advantage is that it is accurate test and does not require distribution hypothesis.We consider applying the permutation test to the test of the high-dimensional two-sample covariance matrix,and construct the LM permutation test,and then use the permutation test to obtain the p-value.The test does not rely on the assumption of normal distribution or the structure of the covariance matrix.(2)Based on another expression of the covariance matrix,this article considers the vectorized covariance test statistic,which is a permutation test for the parameter vector.Therefore,this thesis considers using the vech operator to vectorize the covariance matrix,and uses marginal standard errors instead of covariance.Thus,a permutation test method based on vectorized covariance is constructed.(3)We analyzed the effects of the test method through numerical simulations,mainly verifying the empirical size and empirical power of the test model for the normal distribution and the gamma distribution.We compared the performance of the proposed test method with other existing methods.When the sample is normally distributed,the empirical power of the permutation test and the existing parameter test is not much different.In the case of Gamma distribution,the vectorized covariance test statistic proposed by us is better than the existing test methods.In particular,the covariance matrix permutation test proposed in this paper does not require distribution assumptions,and can handle sparse and dense alternatives. |
