An extension of the Cardioid distributions on circle

Document Type : Original Article

Author

Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

10.22060/ajmc.2020.18285.1029

Abstract

A new family of distributions on the circle is introduced which is a generalization of the Cardioid distributions. The elementary properties such as mean, variance, and the characteristic function are computed. The distribution is shown to be either unimodal or bimodal. The modes are computed. The symmetry of the distribution is characterized. The parameters are shown to be canonic (i.e. uniquely determined by the distribution). This implies that the estimation problem is welldefined. We also show that this new family is a subset of distributions whose Fourier series has degree at most 2 and study the implications of this property. Finally, we study the maximum likelihood estimation for this family.

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References

1] E. A. Yfantis, L. E. Borgman, An extension of the von Mises distribution, Communications in Statistics-Theory and Methods 11 (1982) 1695-1706.
[2] K. V. Mardia, P. E. Jupp, Directional Statistics, Wiley series in probability and statistics. Wiley, Chichester 2000.
[3] K. V. Mardia, Probability and mathematical statistics: statistics of directional data, Academic Press, London 1972.
[4] S. Kato, M. C. Jones, A family of distributions on the circle with links to, and applications arising from, M¨obius transformation, Journal of the American Statistical Association 105 (2010) 249-262
AUT Journal of Mathematics and Computing
Volume 2, Issue 1
Winter and Spring 2021
Pages 45-52

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