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A characterization of the uniform distribution on the circle in the analysis of directional data

Published online by Cambridge University Press:  14 July 2016

M. S. Bingham
M. S. Bingham*
Affiliation:
University of Hull
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Abstract

It was conjectured some time ago by K. V. Mardia that if, for random samples of some fixed size N ≧ 2 from a given non-degenerate circular population, the sample mean direction and the sample resultant length are independently distributed, then the population must be uniformly distributed round the circle. In this paper it is shown that, apart from one minor exception, Mardia's conjecture is true in the case N = 2. No regularity conditions are necessary for the proof. The same problem has been studied, subject to regularity conditions, by Kent, Mardia and Rao (1976) for all sample sizes N ≧ 2 except N = 4.

Keywords

CHARACTERIZATIONCIRCULAR UNIFORM DISTRIBUTIONDIRECTIONAL DATA
Type
Short Communications
Information
Journal of Applied Probability , Volume 15 , Issue 4 , December 1978 , pp. 852 - 857
Copyright
Copyright © Applied Probability Trust 1978 

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References

Mardia, K. V. (1972) Statistics of Directional Data. Academic Press, London.Google Scholar
Kent, J. T., Mardia, K. V. and Rao, J. S. (1976) A characterization of the isotropic distribution. Directional Data Analysis Project, Research Report No. 8, University of Leeds.Google Scholar