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The Cholesky normal distribution for SPD matrices and inference for the mean

Author

Listed:
  • Benoit Ahanda

    (Bradley University)

  • Leif Ellingson

    (Texas Tech University)

  • Daniel E. Osborne

    (Florida Agricultural and Mechanical University)

Abstract

In this paper, a new distribution is introduced in the space of symmetric positive definite (SPD) matrices called the Cholesky normal distribution. Because the space of SPD matrices is a non-Euclidean manifold, standard arithmetic and thus standard statistical methods do not directly apply for data on this space. Instead, researchers typically either perform an intrinsic analysis by defining a Riemannian metric and then projecting the data onto a tangent space or an extrinsic analysis by embedding the space into the space of symmetric matrices. For both approaches, since there are not many distributions defined on the space of SPD matrices, researchers typically use nonparametric inference procedures, which may be too computationally expensive for practical use on large-scale data analyses. Following from Schwartzman (Int Stat Rev 84(3):456–486, 2015), we utilize the Cholesky metric on the space of SPD matrices to define a distribution, investigate some of its properties, establish a relationship between this distribution and the Wishart distribution, and develop parametric inference procedures for the mean of SPD matrices. Lastly, we provide an illustration of the Cholesky mean and covariance using diffusion tensor imaging (DTI) data collected from a cohort of children diagnosed with dyslexia.

Suggested Citation

  • Benoit Ahanda & Leif Ellingson & Daniel E. Osborne, 2025. "The Cholesky normal distribution for SPD matrices and inference for the mean," Statistical Papers, Springer, vol. 66(1), pages 1-17, January.
    • Handle: RePEc:spr:stpapr:v:66:y:2025:i:1:d:10.1007_s00362-024-01640-3
    DOI: 10.1007/s00362-024-01640-3
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    References listed on IDEAS

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    1. Benoit Ahanda & Daniel E. Osborne & Leif Ellingson, 2022. "Robustness of lognormal confidence regions for means of symmetric positive definite matrices when applied to mixtures of lognormal distributions," METRON, Springer;Sapienza Università di Roma, vol. 80(3), pages 281-303, December.
    2. Hendriks, Harrie & Landsman, Zinoviy, 1998. "Mean Location and Sample Mean Location on Manifolds: Asymptotics, Tests, Confidence Regions," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 227-243, November.
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