Covariance Estimation via Fiducial Inference

TL;DR

This paper introduces a fiducial inference method for covariance estimation that provides uncertainty quantification and confidence regions without relying on priors, demonstrating consistency and utility in structure detection.

Contribution

The paper develops a fiducial approach for covariance estimation, establishing its consistency and effectiveness in uncertainty quantification and clique structure identification.

Findings

Fiducial distribution of covariance matrix is consistent.

Samples from the fiducial distribution serve as reliable estimators.

Method enables meaningful confidence regions for covariance matrices.

Abstract

As a classical problem, covariance estimation has drawn much attention from the statistical community for decades. Much work has been done under the frequentist and the Bayesian frameworks. Aiming to quantify the uncertainty of the estimators without having to choose a prior, we have developed a fiducial approach to the estimation of covariance matrix. Built upon the Fiducial Berstein-von Mises Theorem (Sonderegger and Hannig 2014), we show that the fiducial distribution of the covariate matrix is consistent under our framework. Consequently, the samples generated from this fiducial distribution are good estimators to the true covariance matrix, which enable us to define a meaningful confidence region for the covariance matrix. Lastly, we also show that the fiducial approach can be a powerful tool for identifying clique structures in covariance matrices.