Convergence of Sample Eigenvectors of Spiked Population Model

In this paper, for the general non Gaussian spiked population model, where a few fixed eigenvalues of the population covariance matrix are separated from others, we investigate the convergence properties of the eigenvectors of sample covariance matrices corresponding to the spiked population eigenvalues and angle between the population eigenvectors and sample eigenvectors as both the sample size and population size are large.     

Multi-sample test for high-dimensional covariance matrices

In this paper, we propose a new test statistic for testing the equality of high-dimensional covariance matrices for multiple populations. The proposed test statistic generalizes the test of the equality of two population covariance matrices proposed by Li and Chen (2012).     

Reduced-rank estimation of the difference between two covariance matrices

We consider m×m$m\times m$ covariance matrices, _Σ1${\Sigma }_1$ and _Σ2${\Sigma }_2$, which satisfy _Σ2-_Σ1=Δ${\Sigma }_2-{\Sigma }_1=\Delta$, where Δ$\Delta$ has a specified rank. Maximum likelihood estimators of _Σ1${\Sigma }_1$ and _Σ2${\Sigma }_2$ are obtained when sample covariance matrices having Wishart distributions are available and rank(Δ)$\text{rank}(\Delta )$ is known. The likelihood ratio statistic for a test about the value of rank(Δ)$\text{rank}(\Delta )$ is also given and some properties of its null distribution are obtained. The methods developed in this paper are illustrated through an example.     

Crisis cycles in the slightly unstable block random model

ABSTRACT

The paper proposes a new approach for studying the time to time appearing breakdowns in economy. Block random model can describe stability of large complicated systems with variable number of participants. Theoretical background of the model is given by a theorem about the eigenvalues of block random matrices [Juhász F. On the characteristic values of non-symmetric block random matrices. J Theoret Probab. 1990;67:199–205; On the structural eigenvalues of block random matrices. Linear Algebra Appl. 1996;246:225–231]. The model takes into account not only effects of participants but of groups formed from them as well. Slight instability means group level stability and participant level instability [Juhász F. On the turbulence of slightly unstable block random systems. In: Taylor C, et al., editors. Numerical methods for laminar and turbulent flow. Atlanta; 1995. p. 113–121]. Lability index of block random systems is introduced for measuring instability. It is showed that lability index of a slightly unstable block random model is growing while number of participants increases. Alteration in the number of participants makes it possible to describe crisis cycles.     

Markov chain Monte Carlo exact inference for social networks

The uniformly most powerful unbiased test of reciprocity compares the observed number of mutual relations to its exact conditional distribution. Metropolis–Hastings algorithms have been proposed for generating from this distribution in order to perform Monte Carlo exact inference. Triad census statistics are often used to test for the presence of network group structure. We show how one of the proposed Metropolis–Hastings algorithms can be modified to generate from the conditional distribution of the triad census given the in-degrees, the out-degrees and the number of mutual dyads. We compare the results of this algorithm with those obtained by using various approximations.     

模糊线性变换的模糊特征向量

文[l]给出了模糊线性变换的模糊特征向量的定义，在此基础上，本文讨论了模糊特征向量的性质，并给出了某些特殊的模糊线性变换的模糊待征向量的结构。     

Whither jackknifing in stein-rule estimation

In multi-parameter ( multivariate ) estimation, the Stein rule provides minimax and admissible estimators , compromising generally on their unbiasedness. On the other hand, the primary aim of jack-knifing is to reduce the bias of an estimator ( without necessarily compromising on its efficacy ), and, at the same time, jackknifing provides an estimator of the sampling variance of the estimator as well. In shrinkage estimation ( where minimization of a suitably defined risk function is the basic goal ), one may wonder how far the bias-reduction objective of jackknifing incorporates the dual objective of minimaxity ( or admissibility ) and estimating the risk of the estimator ? A critical appraisal of this basic role of jackknifing in shrinkage estimation is made here. Restricted, semi-restricted and the usual versions of jackknifed shrinkage estimates are considered and their performance characteristics are studied . It is shown that for Pitman-type ( local ) alternatives, usually, jackkntfing fails to provide a consistent estimator of the ( asymptotic ) risk of the shrinkage estimator, and a degenerate asymptotic situation arises for the usual fixed alternative case.     

Testing linear hypotheses in high-dimensional regressions

For a multivariate linear model, Wilk's 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 Wilk's Λ including the widely used chi-square approximation. In this paper, we consider necessary modifications to Wilk's 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 Wilk's 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.