LARGE COVARIANCE ESTIMATION THROUGH ELLIPTICAL FACTOR MODELS.

Area of Science:

• Statistics
• Econometrics
• Machine Learning

Background:

• Covariance matrix estimation is crucial in high-dimensional data analysis.
• Existing methods face challenges with large datasets and heavy-tailed distributions.
• Approximate factor models offer a promising structure for simplification.

Purpose of the Study:

• To propose a general Principal Orthogonal Complement Thresholding (POET) framework for large-scale covariance matrix estimation.
• To establish conditions for optimal convergence rates of POET.
• To extend POET to handle heavy-tailed data and conditional graphical models.

Main Methods:

• Utilizing an approximate factor model as the basis for the POET framework.
• Establishing sufficient conditions for optimal convergence rates under various matrix norms.
• Developing a robust estimator using marginal and spatial Kendall's tau for elliptical distributions.
• Investigating conditional graphical models within the POET framework.

Main Results:

• The POET framework achieves optimal convergence rates for covariance matrix estimation.
• The framework provides a transparent recovery of existing results for sub-Gaussian data.
• For the first time, POET exploits conditional sparsity in covariance structures for heavy-tailed data.
• A robust estimator for elliptical distributions is proposed and shown to satisfy the established conditions.

Conclusions:

• The POET framework is a versatile and effective tool for large-scale covariance matrix estimation.
• The theoretical contributions advance the understanding of high-dimensional principal component analysis.
• The proposed methods offer robust solutions for both standard and heavy-tailed data scenarios.