Covariance estimation via fiducial inference

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 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, 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.

Keywords:

• covariance estimation
• sparsity
• fiducial inference
• cliques

2010 Mathematics Subject Classifications:

• Primary 62J10
• 62E20
• 62F25
• Secondary 62F12

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Shi's research was supported in part by the National Library of Medicine Institutional Training Grant T15 LM009451. Hannig's research was supported in part by the National Science Foundation (NSF) under Grant Nos. 1512945, 1633074, and 1916115. Lee's research was supported in part by the NSF under Grant No. 1512945 and 1513484.

Notes on contributors

W. Jenny Shi

W. Jenny Shi obtained her PhD in Statistics from the University of North Carolina. From 2015 to 2018, she was a National Institute of Health postdoctoral fellow at the University of Colorado. She is now a Quantitative Strategist at MassMutual, specializing in financial modeling and strategic initiatives.

Jan Hannig

Jan Hannig received his Mgr (MS equivalent) in mathematics in 1996 from the Charles University, Prague, Czech Republic. He received Ph.D. in statistics and probability in 2000 from Michigan State University under the direction of Professor A.V. Skorokhod. From 2000 to 2008 he was on the faculty of the Department of Statistics at Colorado State University where he was promoted to an Associate Professor. He has joined the Department of Statistics and Operation Research at the University of North Carolina at Chapel Hill in 2008 and was promoted to Professor in 2013. He is an elected member of International Statistical Institute and a fellow of the American Statistical Association and Institute of Mathematical Statistics.

Randy C. S. Lai

Randy C. S. Lai obtained his Ph.D. in Statistics from the University of California, Davis (UC Davis). In 2015–2019 he was an Assistant Professor at the University of Maine. He is now a Visiting Assistant Professor at UC Davis, and he will join Google as a Data Scientist in Spring 2021. His research interests include fiducial inference and statistical computing.

Thomas C. M. Lee

Thomas C. M. Lee is Professor of Statistics and Associate Dean for the Faculty in Mathematical and Physical Sciences at the University of California, Davis. He is an elected Fellow of the American Association for the Advancement of Science (AAAS), the American Statistical Association (ASA), and the Institute of Mathematical Statistics (IMS). From 2013 to 2015 he served as the editor-in-chief for the Journal of Computational and Graphical Statistics, and from 2015 to 2018 he served as the Chair of the Department of Statistics at UC Davis. His recent research interests include astrostatistics, fiducial inference, machine learning, and statistical image and signal processing.