PAGE: Robust Pattern Guided Estimation of Large Covariance Matrix

We study the problem of estimating large covariance matrices under two types of structural assumptions: (i) The covariance matrix is the summation of a low rank matrix and a sparse matrix, and we have some prior information on the sparsity pattern of the sparse matrix; (ii) The data follow a transelliptical distribution. The former structure regulates the parameter space and has its roots in different statistical models (e.g., approximate factor model, spike covariance model, and random effects model) and is motivated by some observations in financial data. The latter structure regulates the data distributions and has its root in copula modeling. Under these assumptions we propose a PAttern Guided Estimation (PAGE) method for estimating the (latent) covariance matrix. The PAGE method is rank based and naturally handles heavy tailed data. Theoretically, we show that: (i) PAGE enjoys the oracle property, i.e., it can recover the latent covariance matrix as if we know the exact low rank structure in advance; (ii) PAGE attains a fast rate of convergence as if the estimation is conducted under the Gaussian distributed data. We further extend PAGE to the situation in which the sparsity pattern is unknown. In this case, our method can be regarded as a robust version of POET in Fan et al. (2013), but is implemented through PAGE. Keyword: Matrix decomposition; Low rank matrix; Sparse matrix; Transelliptical distribution; Sparsity pattern; Catoni’s estimator; Kendall’s tau; Oracle property.