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

Wan Tian, Wenhao Cui, Rui Zhang, Bingyi Jing, Yang Liu, Yijie Peng

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.

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

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.

Paper Structure

This paper contains 16 sections, 5 theorems, 82 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Theoretical Properties
  4. Convergence Analysis of Algorithm \ref{['ADMMalgorithm']}
  5. Non-asymptotic analysis of the IST estimator
  6. Simulation Studies
  7. Baseline Simulation: Covariance Matrix Estimation
  8. Guided Simulation: Toward Empirical Application
  9. Real Data Applications
  10. Conclusion
  11. Proofs for Results
  12. Proof of Lemma \ref{['auxiliarylemma']}
  13. Proof of Theorem \ref{['maintheorem']}
  14. Proof of Theorem \ref{['nasymptoticth1']}
  15. Proof of Theorem \ref{['nasymptoticth2']}
  16. ...and 1 more sections

Key Result

Lemma 3.1

Let $U^* = (\Lambda^*, \Sigma^*)^\top, U^{(q)} = (\Lambda^{(q)}, \Sigma^{(q)})^\top$. Then, we have that the sequence $(\Sigma^{(q)}, \Gamma^{(q)}, \Lambda^{(q)})$ generated by Algorithm ADMMalgorithm satisfies $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Estimation results for DGP1 with an AR(1) covariance structure and $(n,p)=(120,120)$. In the first row, from left to right, are the true covariance matrix, the estimated covariance matrix, and the difference between the two. In the second row, from left to right, are the eigenvalue curves, the variance scatter plot, and the interpolated correlation matrix.
  • Figure 2: The effects of covariance structure, sample size, dimensionality, and parameter on estimation efficiency; the three rows correspond to DGP1, DGP2, and DGP3, respectively.
  • Figure 3: The figure presents the estimated covariance matrix as of October 8, 2024.
  • Figure 4: The figure presents the estimated covariance matrix as of October 9, 2024.

Theorems & Definitions (10)

  • Lemma 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • proof
  • proof
  • proof
  • proof
  • proof