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

TL;DR

This paper introduces a new positive definite covariance matrix estimator for high-dimensional interval-valued data, extending classical methods with a novel soft-thresholding approach and proven convergence.

Contribution

It proposes the Interval-valued Soft-Thresholding (IST) estimator with a positive definiteness constraint, along with an efficient algorithm and theoretical guarantees.

Findings

Simulation studies validate the estimator's effectiveness.

Application to financial data demonstrates practical utility.

The method ensures positive definiteness in high-dimensional settings.

Abstract

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.