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On estimation of a partitioned covariance matrix with linearly structured blocks

Author

Listed:
  • Katarzyna Filipiak

    (Poznan University of Technology)

  • Augustyn Markiewicz

    (Poznan University of Life Sciences)

  • Adam Mieldzioc

    (Poznan University of Life Sciences)

  • Malwina Mrowińska

    (Poznan University of Technology)

Abstract

The aim of this paper is to introduce an estimation method for a linearly structured partitioned covariance matrix. In contrast to well known linear structures of partitioned matrices, for example block compound symmetry, we allow the diagonal blocks of the covariance matrix to be of different dimensions. We adapt the shrinkage method to improve the properties of the projection of the sample covariance matrix onto the linear structure space. As spaces of target matrices, we choose various quadratic subspaces of structure space. This is a novel approach in the context of the structure space under consideration, and as a result a positive definite and well-conditioned estimator having the desired structure is determined. It is also shown that the statistical and algebraic properties of the estimator depend on the choice of target space.

Suggested Citation

  • Katarzyna Filipiak & Augustyn Markiewicz & Adam Mieldzioc & Malwina Mrowińska, 2025. "On estimation of a partitioned covariance matrix with linearly structured blocks," Statistical Papers, Springer, vol. 66(4), pages 1-26, June.
    • Handle: RePEc:spr:stpapr:v:66:y:2025:i:4:d:10.1007_s00362-025-01718-6
    DOI: 10.1007/s00362-025-01718-6
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    References listed on IDEAS

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    1. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
    2. Hannart, Alexis & Naveau, Philippe, 2014. "Estimating high dimensional covariance matrices: A new look at the Gaussian conjugate framework," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 149-162.
    3. Ohlson, Martin & von Rosen, Dietrich, 2010. "Explicit estimators of parameters in the Growth Curve model with linearly structured covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 101(5), pages 1284-1295, May.
    4. Emilie Devijver & Mélina Gallopin, 2018. "Block-Diagonal Covariance Selection for High-Dimensional Gaussian Graphical Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(521), pages 306-314, January.
    5. Ledoit, Olivier & Wolf, Michael, 2021. "Shrinkage estimation of large covariance matrices: Keep it simple, statistician?," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    6. Ikeda, Yuki & Kubokawa, Tatsuya & Srivastava, Muni S., 2016. "Comparison of linear shrinkage estimators of a large covariance matrix in normal and non-normal distributions," Computational Statistics & Data Analysis, Elsevier, vol. 95(C), pages 95-108.
    7. Yifan Yang & Chixiang Chen & Shuo Chen, 2024. "Covariance Matrix Estimation for High-Throughput Biomedical Data with Interconnected Communities," The American Statistician, Taylor & Francis Journals, vol. 78(4), pages 401-411, October.
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