Some Hypotheses Testing For High Dimensional Covariance Matrix And Its Applications

The hypotheses testing of covariance matrix is an important problem of statistical inference in multivariate statistical analysis.It has been applied to various fields,such as image processing,signal processing and genetic genetics.When the data dimension is fixed and sample size is large,the traditional test methods have good effects in the classic large-sample distribution theory.However,with the progress of science and technology,the dimensions and sample size of data are both large.If the traditional test methods are still used,then the results will become very poor.In this paper,we explore the problems of testing the sphericity and identity of high-dimensional population covariance matrices and apply the results to the source number estimation in signal processing.The main work and contributions of this thesis are stated as follows:1.Sphericity test for high-dimensional covariance matrix when the mean of the population is unknownAiming at the spherical test of covariance matrix when the mean of the population is unknown,a test method based on U statistic is proposed in highdimensional case.The U test statistic is quantized as the functional form of linear spectrum statistics of the sample covariance matrix and its asymptotic distribution in the general asymptotic system is obtained by using the random matrix theory,when the population is generally distributed and the mean value is unknown.Numerical simulations demonstrates that the proposed test method has good control over the type-one error probability by comparing with the modified U test,whether the population is normally distributed or not.We also can see that the powers of two test methods are not much difference when the sample size and dimension are both large.But the sample size is small,the test method we proposed has more significant powers.2.Sphericity test of covariance matrix for high-dimensional dataIn the problem of testing for sphericity,many studies rely on the assumption of normal population and are restricted by the classical asymptotic framework.It brings difficulties to the practical applications.We consider the sphericity test in high-dimensional case for the general population.When the dimension and the sample size are both tend to infinity,we mainly study the sphericity test in two cases where the sample size is larger than the dimension and the sample size is fewer than the dimension.For the first case,we construct an LRT-like test statistic and derive the asymptotic distribution of the statistic.Numerical experiments show that the proposed method can not only control the type-one error probability,but also significantly improve the efficiency of the modified LRT.For the second case,we construct two different test statistics by using the first four moments of eigenvalue spectrum distribution of sample covariance matrix,and derive their distribution properties.Numerical experiments show that these two kinds of test statistics can control the type-one error probability,and they are better than the existing test methods in the literature for the different structures of covariance matrix.3.Identity test of covariance matrix for high-dimensional dataA test method based on random matrix theory is proposed for the hypothesis test of whether the covariance matrix is equal to the identity matrix in highdimensional case,when the population is general distribution.According to the inequality relation of the eigenvalue of covariance matrix,a new test statistic is constructed by using the first four moments function of spectrum distribution of the sample eigenvalue,and its asymptotic distribution is obtained in general asymptotic regime.Numerical simulations show that when the population is normal or non-normal distribution,the method proposed can effectively control the type-one error probability.For the spiked strcture of covariance matrix,the test powers of the proposed method have obviously improved by comparing with some existing methods.Through the analysis of practical cases,the results show that the proposed method has more significant effects.4.Source enumeration based on testing for sphericity of high dimensional covariance matrixFor the problem of source enumeration,it is difficult to use the information theoretic criterion when the observation data are non-Gaussian distribution or the sample size is less than the dimension.A method is proposed to estimate the number of the sources in high-dimensional case based on sphericity test.The non-Gaussian observation data are decomposed into signal components and noise components under a presumptive number of source by using unitary transformation.If the noise components do not contain the signal,the corresponding sphericity test statistics will be a certain Gaussian distribution and the number of source can be estimated via generlized Bayesian information criterion(GBIC).The likelihood logarithmic term in the GBIC is represented by the first four order moments of spectral distribution of sample covariance matrix.Thus,we propose an algorithm for estimating the number of sources for the Gaussian and nonGaussian noise based on sphericity test when the sample size and the number of sensors are both tend to infinity.Numerical simulations show that the proposed algorithm has good estimation performance,and has higher detection probability than the compared methods when the sample size is less than the number of sensors.