Inference on the eigenvalues of the covariance matrix of a multivariate normal distribution--geometrical view--

TL;DR

This paper explores the geometric structure of the space of multivariate normal distributions to analyze eigenvalue inference, bias, information loss, and propose new estimators based on differential geometry.

Contribution

It introduces a geometric framework for eigenvalue inference in multivariate normals, analyzing curvature and deriving new estimators from this perspective.

Findings

Analysis of metric and curvature of eigenvalue and eigenvector submanifolds

Quantification of bias in sample eigenvalues

Development of new estimators from geometric insights

Abstract

We consider an inference on the eigenvalues of the covariance matrix of a multivariate normal distribution. The family of multivariate normal distributions with a fixed mean is seen as a Riemannian manifold with Fisher information metric. Two submanifolds naturally arises; one is the submanifold given by fixed eigenvectors of the covariance matrix, the other is the one given by fixed eigenvalues. We analyze the geometrical structures of these manifolds such as metric, embedding curvature under $e$-connection or $m$-connection. Based on these results, we study 1) the bias of the sample eigenvalues, 2)the information loss caused by neglecting the sample eigenvectors, 3)new estimators that are naturally derived from the geometrical view.