Improved Second Order Estimation in the Singular Multivariate Normal Model

TL;DR

This paper introduces novel estimators for covariance, precision matrices, and discriminant coefficients in high-dimensional normal data with low-rank covariance, demonstrating significant empirical improvements over classical methods.

Contribution

It develops unbiased risk estimation-based estimators tailored for low-rank covariance matrices, advancing estimation accuracy in high-dimensional settings.

Findings

Significant empirical performance improvements over classical estimators

Effective estimation of covariance and precision matrices in low-rank scenarios

Enhanced discriminant coefficient estimation in high-dimensional data

Abstract

We consider the problem of estimating covariance and precision matrices, and their associated discriminant coefficients, from normal data when the rank of the covariance matrix is strictly smaller than its dimension and the available sample size. Using unbiased risk estimation, we construct novel estimators by minimizing upper bounds on the difference in risk over several classes. Our proposal estimates are empirically demonstrated to offer substantial improvement over classical approaches.