Inverse Batschelet Distributions for Circular Data
Article first published online: 20 AUG 2011
© 2011, The International Biometric Society
Volume 68, Issue 1, pages 183–193, March 2012
How to Cite
Jones, M. C. and Pewsey, A. (2012), Inverse Batschelet Distributions for Circular Data. Biometrics, 68: 183–193. doi: 10.1111/j.1541-0420.2011.01651.x
- Issue published online: 23 MAR 2012
- Article first published online: 20 AUG 2011
- Received October 2010. Revised May 2011. Accepted June 2011.
- Circular statistics;
- Parameter orthogonality;
- Skew distributions;
- Transformation of scale;
- von Mises distribution
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.