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Robustness of lognormal confidence regions for means of symmetric positive definite matrices when applied to mixtures of lognormal distributions

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  • Benoit Ahanda

    (Bradley University)

  • Daniel E. Osborne

    (Florida Agricultural and Mechanical University)

  • Leif Ellingson

    (Texas Tech University)

Abstract

Symmetric positive definite (SPD) matrices arise in a wide range of applications including diffusion tensor imaging (DTI), cosmic background radiation, and as covariance matrices. A complication when working with such data is that the space of SPD matrices is a manifold, so traditional statistical methods may not be directly applied. However, there are nonparametric procedures based on resampling for statistical inference for such data, but these can be slow and computationally tedious. Schwartzman (Int Stat Rev 84(3):456–486, 2016). introduced a lognormal distribution on the space of SPD matrices, providing a convenient framework for parametric inference on this space. Our goal is to check how robust confidence regions based on this distributional assumption are to a lack of lognormality. The methods are illustrated in a simulation study by examining the coverage probability of various mixtures of distributions.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metron:v:80:y:2022:i:3:d:10.1007_s40300-022-00234-z
    DOI: 10.1007/s40300-022-00234-z
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    References listed on IDEAS

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    1. Armin Schwartzman, 2016. "Lognormal Distributions and Geometric Averages of Symmetric Positive Definite Matrices," International Statistical Review, International Statistical Institute, vol. 84(3), pages 456-486, December.
    2. R. N. Bhattacharya & L. Ellingson & X. Liu & V. Patrangenaru & M. Crane, 2012. "Extrinsic analysis on manifolds is computationally faster than intrinsic analysis with applications to quality control by machine vision," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 28(3), pages 222-235, May.
    3. Osborne, Daniel & Patrangenaru, Vic & Ellingson, Leif & Groisser, David & Schwartzman, Armin, 2013. "Nonparametric two-sample tests on homogeneous Riemannian manifolds, Cholesky decompositions and Diffusion Tensor Image analysis," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 163-175.
    4. 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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