Covariance Matrix Estimation for High-Dimensional Interval-Valued Data with Positive Definiteness

In the realm of high-dimensional data analysis, the estimation of covariance matrices is a fundamental task, and this holds true for interval-valued data as well. However, there is no unified definition for the covariance matrix of interval-valued data, let alone established estimation methods in high-dimensional settings. This paper presents a novel approach to estimating covariance matrices for high-dimensional interval-valued data while ensuring positive definiteness. We begin by assuming that the upper and lower bounds of interval-valued variables share the same dependency structure. Based on this assumption, we extend the classical soft-thresholding covariance matrix estimator to the interval-valued scenario, referred to as the Interval-valued Soft-Thresholding (IST) estimator. Subsequently, to ensure the positive definiteness of the estimator, we impose a positive definiteness constraint on the IST estimator. We derive an alternating direction method to solve the proposed problem and establish its convergence. Under some very mild conditions, we develop a non-asymptotic statistical theory for the proposed estimator. Simulation studies and applications to high-frequency financial data from the CSI 300 Index demonstrated the effectiveness of the proposed estimator.