Tests for large-dimensional shape matrices via Tyler's M estimators.

Area of Science:

• Statistics
• Multivariate Analysis
• Robust Statistics

Background:

• Classical Tyler's M estimator theory is limited to low-dimensional data and elliptical populations.
• High-dimensional statistical inference for Tyler's M estimator in general populations is not well-established.

Purpose of the Study:

• Develop hypothesis tests for identity and equality of shape matrices in a large-dimensional setting.
• Extend the applicability of Tyler's M estimator to scenarios where dimension $p$ approaches sample size $n$.
• Provide a unified spectral theory for large-dimensional Tyler's M estimators for general populations.

Main Methods:

• Utilizing eigenvalues of Tyler's M estimator.
• Developing tests for identity and equality of shape matrices.
• Establishing a unified theory for the spectrum of large-dimensional Tyler's M estimators.

Main Results:

• Proposed tests are effective in a large-dimensional framework ($p/n$ converges to a limit in (0,1)).
• Tests are applicable to a wide range of multivariate distributions, including elliptical ones.
• Simulation studies confirm the good performance and robustness of the developed tests.

Conclusions:

• The study successfully extends Tyler's M estimator and hypothesis testing to high-dimensional settings.
• The developed methods offer a robust alternative to traditional covariance matrix analysis.
• Empirical analysis demonstrates the practical utility in portfolio analysis.