Regularized estimation of the Mahalanobis distance based on modified Cholesky decomposition

Abstract

Estimating inverse covariance matrix is an essential part of many statistical methods. This paper proposes a regularized estimator for the inverse covariance matrix. Modified Cholesky decomposition (MCD) is utilized to construct positive definite estimators. Instead of directly regularizing the inverse covariance matrix itself, we impose regularization on the Cholesky factor. The estimated inverse covariance matrix is used to build Mahalanobis distance (MD). The proposed method is evaluated by detecting outliers through simulations and empirical studies.

Keywords:

• Modified Cholesky decomposition
• Mahalanobis distance
• regularization
• smoothing
• outlier detection

Acknowledgments

Deliang Dai acknowledges the warm hospitality of Department of Mathematics at The university of Manchester where this work was initiated, during a visit funded by Linnaeus University. Authors are grateful to the referees for their detailed review and helpful suggestions which has resulted in significant improvement of this manuscript.