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A possibly asymmetric multivariate generalization of the Möbius distribution for directional data

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  • Uesu, Kagumi
  • Shimizu, Kunio
  • SenGupta, Ashis

Abstract

A family of possibly asymmetric distributions on the unit hyper-disc with center at the origin is proposed. The paper presents a non-trivial multivariate generalization of the Möbius distribution on the unit disc. The family is obtained by applying a conformal mapping to the spherically symmetric beta distribution. The density functions of the family are unimodal, monotonic or uniantimodal. The conditional distribution of direction cosine given the length is a t-distribution on the sphere. The conditional distribution of the length given the direction cosine has a simple closed form expression, though not of any standard known distribution. Modality, skewness and direction parameters are globally orthogonal in the sense that the Fisher information matrix is diagonal. The proposed model on the hyper-disc, introducing this probability distribution for the very first time, is applied to an emerging area of astrophysics for a dataset on gamma-ray bursts and to a challenging area of geoinformatics for a dataset on worldwide earthquakes with magnitude greater than or equal to 7.0MW.

Suggested Citation

  • Uesu, Kagumi & Shimizu, Kunio & SenGupta, Ashis, 2015. "A possibly asymmetric multivariate generalization of the Möbius distribution for directional data," Journal of Multivariate Analysis, Elsevier, vol. 134(C), pages 146-162.
    • Handle: RePEc:eee:jmvana:v:134:y:2015:i:c:p:146-162
    DOI: 10.1016/j.jmva.2014.11.004
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    References listed on IDEAS

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    1. M. Jones, 2004. "The Möbius distribution on the disc," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(4), pages 733-742, December.
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    4. Seshadri, V., 1991. "A family of distributions related to the McCullagh family," Statistics & Probability Letters, Elsevier, vol. 12(5), pages 373-378, November.
    5. Jones, M.C. & Pewsey, Arthur, 2005. "A Family of Symmetric Distributions on the Circle," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1422-1428, December.
    6. Jones, M. C., 2002. "Marginal Replacement in Multivariate Densities, with Application to Skewing Spherically Symmetric Distributions," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 85-99, April.
    7. Grace Shieh & Richard Johnson, 2005. "Inferences based on a bivariate distribution with von Mises marginals," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(4), pages 789-802, December.
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