Equivariant minimax dominators of the MLE in the array normal model

Area of Science:

• Statistics
• Multivariate Analysis
• Data Science

Background:

• The array normal model is crucial for dependency inference in multiway data arrays, assuming separable covariance matrices.
• Existing inference methods (Maximum Likelihood, Bayesian) lack established optimality properties.
• Optimal estimation for array normal models remains an open research area.

Purpose of the Study:

• To derive and analyze estimators with proven optimality properties for the array normal model.
• To compare the performance of novel equivariant estimators against the Maximum Likelihood Estimator (MLE).
• To investigate risk properties and potential improvements for covariance estimation in multiway data.

Main Methods:

• Development of a uniformly minimum risk equivariant estimator (UMREE) using a generalized Bayes procedure under a lower triangular product group.
• Analysis of minimax properties and dominance of the UMREE over the MLE.
• Introduction of an orthogonally equivariant modification to further improve estimator risk.

Main Results:

• A UMREE was successfully derived for the array normal model, demonstrating minimax properties.
• The derived UMREE was shown to dominate the MLE.
• An orthogonally equivariant modification further reduced risks compared to the UMREE and MLE.
• Numerical comparisons revealed substantial risk reductions for equivariant estimators over the MLE.

Conclusions:

• Equivariant estimation methods offer significant advantages for the array normal model, particularly in risk reduction.
• The study provides a theoretical framework and practical demonstration of superior estimators for multiway data dependency analysis.
• These findings advance the field of multivariate statistics and statistical inference for complex data structures.