Inverse Batschelet Distributions for Circular Data

Authors

  • M. C. Jones,

    Corresponding author
    1. Department of Mathematics & Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, U.K.
      email: m.c.jones@open.ac.uk
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  • Arthur Pewsey

    Corresponding author
    1. Department of Mathematics, Escuela Politécnica, University of Extremadura, 10003 Cáceres, Spain
      email: apewsey@unex.es
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email:m.c.jones@open.ac.uk

email:apewsey@unex.es

Abstract

Summary We provide four-parameter families of distributions on the circle which are unimodal and display the widest ranges of both skewness and peakedness yet available. Our approach is to transform the scale of a generating distribution, such as the von Mises, using various nontrivial extensions of an approach first used in Batschelet’s (1981, Circular Statistics in Biology) book. The key is to employ inverses of Batschelet-type transformations in certain ways; these exhibit considerable advantages over direct Batschelet transformations. The skewness transformation is especially appealing as it has no effect on the normalizing constant. As well as a variety of interesting theoretical properties, when likelihood inference is explored these distributions display orthogonality between elements of a pairing of parameters into (location, skewness) and (concentration, peakedness). Further, the location parameter can sometimes be made approximately orthogonal to all the other parameters. Profile likelihoods come to the fore in practice. Two illustrative applications, one concerning the locomotion of a Drosophila fly larva, the other analyzing a large set of sudden infant death syndrome data, are investigated.

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