Ultrahigh Dimensional Precision Matrix Estimation via Refitted Cross Validation.

Area of Science:

• Statistics
• Econometrics
• Machine Learning

Background:

• Estimating sparse precision matrices is crucial in high-dimensional data analysis.
• Existing regularization methods may struggle with ultrahigh dimensional data due to spurious correlations.
• The Gaussian assumption is often restrictive in real-world applications.

Purpose of the Study:

• To develop a novel and robust estimation procedure for ultrahigh dimensional sparse precision matrices.
• To address limitations of existing methods, particularly concerning spurious correlations and Gaussian assumptions.
• To provide a method easily implementable with existing statistical software.

Main Methods:

• Proposes a refitted cross-validation (RCV) method based on the Cholesky decomposition of the precision matrix.
• The RCV method does not require the Gaussian assumption.
• Leverages existing software for ultrahigh dimensional linear regression for implementation.

Main Results:

• Establishes the consistency of the RCV estimation procedure.
• Demonstrates that RCV achieves the same convergence rate as methods assuming banded structures, without such assumptions.
• Monte Carlo simulations confirm the superior finite sample performance of RCV compared to existing methods.

Conclusions:

• The proposed RCV method offers a consistent and efficient approach for sparse precision matrix estimation in ultrahigh dimensions.
• RCV outperforms existing methods, particularly in scenarios prone to spurious correlations.
• The method's applicability is demonstrated through an empirical analysis in asset allocation.