Statistical analysis of latent generalized correlation matrix estimation in transelliptical distribution

Area of Science:

• Statistics
• Multivariate Analysis
• Robust Statistics

Background:

• Correlation matrices are crucial for multivariate methods like graphical model estimation and factor analysis.
• Pearson's sample correlation matrix is standard but performs poorly with heavy-tailed distributions and outliers.
• Robust alternatives are needed for reliable correlation matrix estimation in diverse data scenarios.

Purpose of the Study:

• To investigate the theoretical properties of Kendall's tau sample correlation matrix for estimating population and latent Pearson's correlation matrices.
• To analyze the performance of a transformed Kendall's tau estimator under spectral and restricted spectral norms.
• To establish conditions for optimal convergence rates in high-dimensional correlation matrix estimation.

Main Methods:

• Analysis of Kendall's tau sample correlation matrix and its transformed version.
• Theoretical investigation under spectral and restricted spectral norms.
• Introduction of the 'sign subgaussian condition' for rank-based estimators.

Main Results:

• The study quantifies convergence rates using 'effective rank' for the spectral norm.
• A novel 'sign subgaussian condition' is presented for optimal convergence under the restricted spectral norm.
• The proposed methods achieve optimal convergence rates without requiring moment conditions.

Conclusions:

• Kendall's tau based correlation matrix estimation offers a robust and theoretically sound alternative to Pearson's method, especially for non-Gaussian data.
• The findings provide theoretical guarantees for high-dimensional correlation matrix estimation under specific conditions.
• This research advances robust statistical methods for analyzing complex datasets with potential outliers or heavy tails.