Statistica Sinica 27 (2017), 1299-1317
Abstract: Estimation of large precision matrices is fundamental to high-dimensional inference. An important issue is to deal with ill-conditioning of the precision matrix estimate, typically encountered in finite-samples, but rarely studied in the iterature. In this paper, we focus on estimating the precision matrix by imposing a bound on the condition number of the estimate, which effectively ensures wellconditioning. Specifically, we propose a correlation-based estimator, constrained with both the condition number and the 𝐿1 penalty, yielding a precision matrix estimator with theoretically guaranteed rate of convergence. This result further enables us to demonstrate that incorporating the 𝐿1 penalty is necessary for achieving consistency of the resulting estimator in typical high-dimensional settings, while inconsistency will occur when the 𝐿1 penalty is absent. An algorithm based on the alternating direction method of multipliers is developed to implement the proposed method, which reveals the satisfactory performance in simulation studies. An application of the method to a call center data is illustrated.
Key words and phrases: Condition number, covariance matrix, penalization, precision matrix, sparsity.