Applications Of Spectral Analysis Of Large Dimensional Random Matrices In Multivariate Statistical Analysis

For decades, with the rapid development of computer technology, large di-mensional data analysis plays more and more important role in modern scientific research. Especially, in the fields of microarray data in biology, stock market anal-ysis in finance and wireless communication networks,etc. Unfortunately, the tradi-tional statistical tools can not catch up the development of the data analysis. The basic problem is that the limiting theory of traditional statistics typically assume a large sample size n with respect to the number of variables p, and bring seriously poor results with high dimension. To compensate the effects due to large p, this dissertation proposes some new statistical methods for this high-dimensional data setting, which would be based on asymptotic theory that both n and p approach infinity.In this dissertation, we review two aspects of multivariate analysis:covariance matrices and mean vectors. Based on the random matrix theory(RMT), we first give an explanation to the failure of traditional likelihood ratio procedures for test-ing about covariance matrices or mean vectors from Gaussian populations when the dimension is large compared to the sample size. Next, using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random F-matrices, we present necessary corrections for these LR tests to cope with high-dimensional effects:testing a covariance matrix equals to a given matrix; testing equality of two covariance matrices; testing on coefficients in a linear regression model; testing equality of means of several populations with common covariance. Furthermore, consider a special case of testing the equality between two covariance matrices, we show that the proposed correction applies equally for non-Gaussian populations yielding a valid pseudo-likelihood ratio test. The asymptotic distribu-tions of these corrected tests under the null are given. Simulations demonstrate that the corrected LR tests yield a realized size close to nominal level for both moderate p (around 20) and high dimension, while the traditional LR tests fail. It shows that although the large dimensional corrected LR tests are based on the theory that both sample size and dimension approach infinite, they have robustness on the dimension p and are feasible in practice.