Symmetric Unimodal Models for Directional Data Motivated by Inverse Stereographic Projection

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  • Abe Toshihiro
    School of Fundamental Science and Technology, Keio University, Yokohama, Japan.
  • Shimizu Kunio
    Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan.
  • Pewsey Arthur
    Mathematics Department, Escuela Politécnica, University of Extremadura, 10003 Cáceres, Spain.

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In this paper, a modified inverse stereographic projection, from the real line to the circle, is used as the motivation for a means of resolving a discontinuity in the Minh–Farnum family of circular distributions. A four-parameter family of symmetric unimodal distributions which extends both the Minh–Farnum and Jones–Pewsey families is proposed. The normalizing constant of the density can be expressed in terms of Appell's function or, equivalently, the Gauss hypergeometric function. Important special cases of the family are identified, expressions for its trigonometric moments are obtained, and methods for simulating random variates from it are described. Parameter estimation based on method of moments and maximum likelihood techniques is discussed, and the latter approach is used to fit the family of distributions to an illustrative data set. A further extension to a family of rotationally symmetric distributions on the sphere is briefly made.

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