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Measures of goodness of fit obtained by almost-canonical transformations on Riemannian manifolds

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  • Jupp, P.E.
  • Kume, A.

Abstract

The standard method of transforming a continuous distribution on the line to the uniform distribution on [0,1] is the probability integral transform. Analogous transforms exist on compact Riemannian manifolds, X, in that for each distribution with continuous positive density on X, there is a continuous mapping of X to itself that transforms the distribution into the uniform distribution. In general, this mapping is far from unique. This paper introduces the construction of an almost-canonical version of such a probability integral transform. The construction is extended to shape spaces, Cartan–Hadamard manifolds, and simplices.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:jmvana:v:176:y:2020:i:c:s0047259x19300211
    DOI: 10.1016/j.jmva.2019.104579
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    References listed on IDEAS

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    1. M. Jones & Arthur Pewsey & Shogo Kato, 2015. "On a class of circulas: copulas for circular distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(5), pages 843-862, October.
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    3. Jupp, P.E., 2015. "Copulae on products of compact Riemannian manifolds," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 92-98.
    4. Boulerice, Bernard & Ducharme, Gilles R., 1997. "Smooth Tests of Goodness-of-Fit for Directional and Axial Data," Journal of Multivariate Analysis, Elsevier, vol. 60(1), pages 154-175, January.
    5. K. V. Mardia, 1999. "Directional statistics and shape analysis," Journal of Applied Statistics, Taylor & Francis Journals, vol. 26(8), pages 949-957.
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    Cited by:

    1. Fernández de Marcos Giménez de los Galanes, Alberto, 2022. "Data-driven stabilizations of goodness-of-fit tests," DES - Working Papers. Statistics and Econometrics. WS 35324, Universidad Carlos III de Madrid. Departamento de Estadística.
    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. Fernández-de-Marcos, Alberto & García-Portugués, Eduardo, 2023. "Data-driven stabilizations of goodness-of-fit tests," Computational Statistics & Data Analysis, Elsevier, vol. 179(C).
    4. 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.

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