Improved Calibration of High-Dimensional Precision Matrices

Estimation of a precision matrix (i.e., the inverse covariance matrix) is a fundamental problem in statistical signal processing applications. When the observation dimension is of the same order of magnitude as the number of samples, the conventional estimators of covariance matrix and its inverse perform poorly. In order to obtain well-behaved estimators in high-dimensional settings, we consider a general class of estimators of covariance matrices and precision matrices based on weighted sampling and linear shrinkage. The estimation error is measured in terms of both quadratic loss and Stein´s loss, and these loss functions are used to calibrate the set of parameters defining our proposed estimator. In an asymptotic setting where the observation dimension is comparable in magnitude to the number of samples, we provide estimators of the precision matrix that are as good as their oracle counterparts. We test our estimators with both synthetic data and financial market data, and Monte Carlo simulations show the advantage of our precision matrix estimator compared with well known estimators in finite sample size settings.