Shurong Zheng

Corrections to LRT on large-dimensional covariance matrix by RMT

In this paper, we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from Gaussian populations when the dimension p is large compared to the sample size n. Next, using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random F -matrices, we propose necessary corrections for these LR tests to cope with high-dimensional effects. The asymptotic distributions 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 with χ 2 approximation fails. Another contribution from the paper is that for testing the equality between two covariance matrices, the proposed correction applies equally for non-Gaussian populations yielding a valid pseudo-likelihood ratio test.

Substitution Principle for CLT of Linear Spectral Statistics of High-Dimensional Sample Covariance Matrices with Applications to Hypothesis Testing

Abstract. Sample covariance matrices are widely used in multivariate statistical analysis. The central limit theorems (CLT’s) for linear spectral statistics of high-dimensional non-centralized sample covariance matrices have received considerable attention in random matrix theory and have been applied to many high-dimensional statistical problems. However, known population mean vectors are assumed for non-centralized sample covariance matrices, some of which even assume Gaussian-like moment conditions. In fact, there are still another two most frequently used sample covariance matrices: the ME (moment estimator, constructed by subtracting the sample mean vector from each sample vector) and the unbiased sample covariance matrix (by changing the denominator n as N = n − 1 in the ME) without depending on unknown population mean vectors. In this paper, we not only establish the new CLT’s for non-centralized sample covariance matrices when the Gaussian-like moment conditions do not hold but also characterize the non-negligible differences among the CLT’s for the three classes of high-dimensional sample covariance matrices by establishing a substitution principle: by substituting the adjusted sample size N = n − 1 for the actual sample size n in the centering term of the new CLT’s, we obtain the CLT of the unbiased sample covariance matrices. Moreover, it is found that the difference between the CLT’s for the ME and unbiased sample covariance matrix is non-negligible in the centering term although the only difference between two sample covariance matrices is a normalization by n and n− 1, respectively. The new results are applied to two testing problems for high-dimensional covariance matrices.

Testing linear hypotheses in high-dimensional regressions

For a multivariate linear model, Wilks likelihood ratio test (LRT) constitutes one of the cornerstone tools. However, the computation of its quantiles under the null or the alternative hypothesis requires complex analytic approximations, and more importantly, these distributional approximations are feasible only for moderate dimension of the dependent variable, say p≤20. On the other hand, assuming that the data dimension p as well as the number q of regression variables are fixed while the sample size n grows, several asymptotic approximations are proposed in the literature for Wilks Λ including the widely used chi-square approximation. In this paper, we consider necessary modifications to Wilks test in a high-dimensional context, specifically assuming a high data dimension p and a large sample size n. Based on recent random matrix theory, the correction we propose to Wilks test is asymptotically Gaussian under the null hypothesis and simulations demonstrate that the corrected LRT has very satisfactory size and power, surely in the large p and large n context, but also for moderately large data dimensions such as p=30 or p=50. As a byproduct, we give a reason explaining why the standard chi-square approximation fails for high-dimensional data. We also introduce a new procedure for the classical multiple sample significance test in multivariate analysis of variance which is valid for high-dimensional data.

CLT for linear spectral statistics of a rescaled sample precision matrix

Central limit theorems (CLTs) of linear spectral statistics (LSS) of general Fisher matrices F are widely used in multivariate statistical analysis where F = SyMSx−1M∗ with a deterministic complex matrix M and two sample covariance matrices Sx and Sy from two independent samples with sample sizes m and n. As the first step to obtain the CLT, it is necessary to establish the CLT for LSS of the random matrix MSx−1M∗, or equivalently that of Sx−1T, that is a sample precision matrix rescaled by a general non-negative definite Hermitian matrix T = M∗M. Because the scaling matrix T in many large-dimensional problems may not be invertible, the result does not simply follow from the celebrated CLT by Bai and Silverstein (2004). Thus, we have to alternatively derive the CLT of LSS of Sx−1T where the inverse of T may not exist, thus extending Bai and Silverstein’s CLT. As a further innovation of the paper, general populations for the sample covariance matrix Sx are covered requiring the existence a fourth-order moment of arbitrary value, that is not necessarily matching the values of the Gaussian case.

A new test of multivariate nonlinear causality

The multivariate nonlinear Granger causality developed by Bai et al. (2010) (Mathematics and Computers in simulation. 2010; 81: 5-17) plays an important role in detecting the dynamic interrelationships between two groups of variables. Following the idea of Hiemstra-Jones (HJ) test proposed by Hiemstra and Jones (1994) (Journal of Finance. 1994; 49(5): 1639-1664), they attempt to establish a central limit theorem (CLT) of their test statistic by applying the asymptotical property of multivariate U-statistic. However, Bai et al. (2016) (2016; arXiv: 1701.03992) revisit the HJ test and find that the test statistic given by HJ is NOT a function of U-statistics which implies that the CLT neither proposed by Hiemstra and Jones (1994) nor the one extended by Bai et al. (2010) is valid for statistical inference. In this paper, we re-estimate the probabilities and reestablish the CLT of the new test statistic. Numerical simulation shows that our new estimates are consistent and our new test performs decent size and power.

CLT for eigenvalue statistics of large-dimensional general Fisher matrices with applications

Random Fisher matrices arise naturally in multivariate statistical analysis and understanding the properties of its eigenvalues is of primary importance for many hypothesis testing problems like testing the equality between two covariance matrices, or testing the independence between sub-groups of a multivariate random vector. Most of the existing work on random Fisher matrices deals with a particular situation where the population covariance matrices are equal. In this paper, we consider general Fisher matrices with arbitrary population covariance matrices and develop their spectral properties when the dimensions are proportionally large compared to the sample size. The paper has two main contributions: first the limiting distribution of the eigenvalues of a general Fisher matrix is found and second, a central limit theorem is established for a wide class of functionals of these eigenvalues. Applications of the main results are also developed for testing hypotheses on high-dimensional covariance matrices.