Covariance Matrices, Precision (Concentration) Matrices, Estimation, and Tests

Wei Lan and Chih-Ling Tsai
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Wei Lan: Southwestern University of Finance and Economics, School of Statistics and Data Science and Center of Statistical Research
Chih-Ling Tsai: University of California - Davis, Graduate School of Management

Chapter Chapter 2 in Covariance Analysis and Beyond, 2026, pp 17-35 from Springer

Abstract: Abstract This chapter introduces covariance matrices and precision (concentration) matrices. We then demonstrate the application of covariance matrices for obtaining the mean, variance, and covariance of quadratic forms. In addition, four types of covariance estimation (moment estimationMoment estimation, maximum likelihood estimationMaximum likelihood estimator (MLE), penalized estimationPenalized estimation, and robust estimationRobust estimation) are presented. Afterward, covariance estimations via eigenvector thresholdingEigenvector thresholding and eigenvalue shrinkageEigenvalue shrinkage are explored. Subsequently, precision matrixPrecision matrix estimationPrecision matrix, time-dependent covariance matrices, positive definitenessPositive definiteness, and the tests of covariance matrices are studied. Finally, empirical examples of the applications of covariance matrices are briefly discussed.

Date: 2026
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DOI: 10.1007/978-3-032-08796-6_2

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