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Copulae on products of compact Riemannian manifolds

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  • Jupp, P.E.

Abstract

One standard way of considering a probability distribution on the unit n-cube, [0,1]n, due to Sklar (1959), is to decompose it into its marginal distributions and a copula, i.e. a probability distribution on [0,1]n with uniform marginals. The definition of copula was extended by Jones et al. (2014) to probability distributions on products of circles. This paper defines a copula as a probability distribution on a product of compact Riemannian manifolds that has uniform marginals. Basic properties of such copulae are established. Two fairly general constructions of copulae on products of compact homogeneous manifolds are given; one is based on convolution in the isometry group, the other using equivariant functions from compact Riemannian manifolds to their spaces of square integrable functions. Examples illustrate the use of copulae to analyse bivariate spherical data and bivariate rotational data.

Suggested Citation

  • Jupp, P.E., 2015. "Copulae on products of compact Riemannian manifolds," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 92-98.
    • Handle: RePEc:eee:jmvana:v:140:y:2015:i:c:p:92-98
    DOI: 10.1016/j.jmva.2015.04.008
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    References listed on IDEAS

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    1. Scarsini, Marco, 1989. "Copulae of probability measures on product spaces," Journal of Multivariate Analysis, Elsevier, vol. 31(2), pages 201-219, November.
    2. Jupp, P. E., 2001. "Modifications of the Rayleigh and Bingham Tests for Uniformity of Directions," Journal of Multivariate Analysis, Elsevier, vol. 77(1), pages 1-20, April.
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    Cited by:

    1. Davy Paindaveine & Thomas Verdebout, 2019. "Inference for Spherical Location under High Concentration," Working Papers ECARES 2019-02, ULB -- Universite Libre de Bruxelles.
    2. Arthur Pewsey & Eduardo García-Portugués, 2021. "Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 1-58, March.
    3. Shogo Kato & Arthur Pewsey & M. C. Jones, 2022. "Tractable circula densities from Fourier series," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(3), pages 595-618, September.
    4. Jupp, P.E. & Kume, A., 2020. "Measures of goodness of fit obtained by almost-canonical transformations on Riemannian manifolds," Journal of Multivariate Analysis, Elsevier, vol. 176(C).

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