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

Benoit Ahanda (), Leif Ellingson () and Daniel E. Osborne ()
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Benoit Ahanda: Bradley University
Leif Ellingson: Texas Tech University
Daniel E. Osborne: Florida Agricultural and Mechanical University

Statistical Papers, 2025, vol. 66, issue 1, No 19, 17 pages

Abstract: 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.

Keywords: SPD Matrices; Cholesky normal; Confidence regions; Lower triangular matrices; Wishart distribution; Cholesky distance for SPD matrices (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s00362-024-01640-3

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