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This paper investigates the central limit theorem for linear spectral statistics of high dimensional sample covariance matrices of the form $${\mathbf {B}}_n=n^{-1}\sum _{j=1}^{n}{\mathbf {Q}}{\mathbf {x}}_j{\mathbf {x}}_j^{*}{\mathbf {Q}}^{*}$$ under the assumption that $$p/n\rightarrow y>0$$ , where $${\mathbf {Q}}$$ is a $$p\times k$$ nonrandom matrix and $$\{{\mathbf {x}}_j\}_{j=1}^n$$ is a sequence of independent k-dimensional random vector with independent entries. A key novelty here is that the dimension $$k\ge p$$ can be arbitrary, possibly infinity. This new model of sample covariance matrix $${\mathbf {B}}_n$$ covers most of the known models as its special cases. For example, standard sample covariance matrices are obtained with $$k=p$$ and $${\mathbf {Q}}={\mathbf {T}}_n^{1/2}$$ for some positive definite Hermitian matrix $${\mathbf {T}}_n$$ . Also with $$k=\infty $$ our model covers the case of repeated linear processes considered in recent high-dimensional time series literature. The CLT found in this paper substantially generalizes the seminal CLT in Bai and Silverstein (Ann Probab 32(1):553–605, 2004). Applications of this new CLT are proposed for testing the AR(1) or AR(2) structure for a causal process. Our proposed tests are then used to analyze a large fMRI data set.

In this paper, we propose a new test on the linear combinations of covariance matrices in high-dimensional data. Our statistic can be applied to many hypothesis tests on covariance matrices. In particular, the test proposed by Li and Chen (Ann Stat 40:908–940, 2012) on the homogeneity of two population covariance matrices is a special case of our test. The results are illustrated by an empirical example in financial portfolio allocation.

In this paper, we adopt the eigenvector empirical spectral distribution (VESD) to investigate the limiting behavior of eigenvectors of a large dimensional Wigner matrix $$\mathbf {W}_n.$$ In particular, we derive the optimal bound for the rate of convergence of the expected VESD of $$\mathbf{W}_n$$ to the semicircle law, which is of order $$O(n^{-1/2})$$ under the assumption of having finite 10th moment. We further show that the convergence rates in probability and almost surely of the VESD are $$O(n^{-1/4})$$ and $$O(n^{-1/6}),$$ respectively, under finite eighth moment condition. Numerical studies demonstrate that the convergence rate does not depend on the choice of unit vector involved in the VESD function, and the best possible bound for the rate of convergence of the VESD is of order $$O(n^{-1/2}).$$

In order to investigate property of the eigenvector matrix of sample covariance matrix $$\mathbf {S}_n$$ , in this paper, we establish the central limit theorem of linear spectral statistics associated with a new form of empirical spectral distribution $$H^{\mathbf {S}_n}$$ , based on eigenvectors and eigenvalues of sample covariance matrix $$\mathbf {S}_n$$ . Using Bernstein polynomial approximations, we prove the central limit theorem for linear spectral statistics of $$H^{\mathbf {S}_n}$$ , indexed by a set of functions with continuous third order derivatives over an interval including the support of Marcenko–Pastur law. This result provides further evidences to support the conjecture that the eigenmatrix of sample covariance matrix is asymptotically Haar distributed.

In this paper we investigate the Fisher information matrix of a rounded ranked set sampling (RSS) sample and show that the sample is always more informative than a rounded simple random sampling (SRS) sample of the same size. On the other hand, we propose a new method to approximate maximum likelihood estimates (MLE) of unknown parameters for this model and further establish the strong consistency and asymptotic normality of the proposed estimators. Simulation experiments show that the approximated MLE based on rounded RSS is always more efficient than those based on rounded SRS.