Random Matrices: Theory and Applications

In this paper, we derive asymptotic null and non-null distributions of three test statistics, namely, the likelihood ratio criterion, the Lawley–Hotelling criterion, and the Bartlett–Nanda–Pillai criterion, for tests of redundancy in high-dimensional (HD) canonical correlation analysis. Since our setting is that the dimension of one of two observation vectors may be large but does not exceed the sample size, we use a HD asymptotic framework such that the sample size and the dimension divided by the sample size tend to $\infty$ and a positive constant within [0,1), respectively, for evaluating asymptotic distributions. Through simulation experiments, it is shown that our proposed HD approximations are more accurate than those under the classical asymptotic framework, i.e. only the sample size tends to $\infty$.

We derive new expressions for the densities of extreme eigenvalues of a Beta–MANOVA matrix, which generalize the classical result of Khatri for the real case. We also present new expressions for the distributions of those eigenvalues which are valid for any values of the parameters.

In this paper, we will propose two new estimators for sparse covariance matrix. Our starting point is to make the estimator of each element of covariance matrix more robust. More precisely, we will trim the observations for each pairwise product of components of population as a first step. Then we form the sample covariance matrices based on the trimmed data. Finally, we apply the thresholding to the derived sample covariance matrices. These two new estimators will be shown to achieve the optimal convergence rate.

We study the recurrence coefficients of the monic polynomials $P_n(z)$ orthogonal with respect to the deformed (also called semi-classical) Freud weight