Indian Journal of Pure and Applied Mathematics

, Volume 50, Issue 4, pp 953–969 | Cite as

Skew-symmetric circular distributions and their structural properties

  • Mojtaba HatamiEmail author
  • Mohammad Hossein AlamatsazEmail author


A general method of constructing a circular model is to apply transformation to its argument. In this paper, we propose a new transformation in order to construct a large class of new skew-symmetric circular models. We consider a symmetric unimodal distribution as our base model and provide general results for the modality, skewness and shape properties of the resulting circular distributions. The new class enfolds two well-known and common classes of unimodal symmetric and asymmetric circular distributions. The skewness parameter in the new class greatly improves the flexibility of our model to cover symmetric, asymmetric, unimodal and multimodal data. Further, we consider Wrapped Cauchy distribution as a special case for our base distribution and introduce the Skew-Symmetric Wrapped Cauchy (SSWC) distribution, discuss its sub-models and estimate its parameters by the maximum likelihood method. Finally, an application of SSWC distribution and its inferential methods are illustrated using two real data examples.

Key words

Circular statistics argument transformation unimodality peakedness skewness wrapped Cauchy distribution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The first author is grateful to the Graduate Office of the University of Isfahan, Iran for their support.


  1. 1.
    T. Abe and A. Pewsey, Sine-skewed circular distributions, Statist. Papers, 52 (2011), 683–707.MathSciNetCrossRefGoogle Scholar
  2. 2.
    T. Abe, A. Pewsey, and K. Shimizu, Extending circular distributions through transformation of argument, Ann. Inst. Statist. Math., 65 (2013), 833–858.MathSciNetCrossRefGoogle Scholar
  3. 3.
    T. Abe, K. Shimizu, and A. Pewsey, On Papakonstantinous extension of the cardioid distribution, Statist. Probab. Lett., 79 (2009), 2138–2147.MathSciNetCrossRefGoogle Scholar
  4. 4.
    E. Batschelet, Circular statistics in biology, Academic Press, London, 1981.zbMATHGoogle Scholar
  5. 5.
    B. V. Bishop, The frequency of thunderstorms at Kew observatory, Met. Mag., 76 (1947), 108–111.Google Scholar
  6. 6.
    J. J. Fernández-Durán, Circular distributions based on nonnegative trigonometric sums, Biometrics, 60 (2004), 499–503.MathSciNetCrossRefGoogle Scholar
  7. 7.
    E. Hernández-Sánchez and B. Scarpa, A wrapped flexible generalized skewnormal model for a bimodal circular distribution of wind directions, Chil. J. Statist., 3 (2012), 131–143.Google Scholar
  8. 8.
    R. Gatto and S. R. Jammalamadaka, The generalized von Mises distribution, Stat. Methodol., 4 (2007), 341–353.MathSciNetCrossRefGoogle Scholar
  9. 9.
    I. S. Gradshteyn and I. M. Ryzhik, Tables of integrals, series, and products, 7nd edn, Academic Press, San Diego, 2007.zbMATHGoogle Scholar
  10. 10.
    S. R. Jammalamadaka and T. J. Kozubowski, New families of wrapped distributions for modeling skew circular data, Comm. Statist. Theory Methods, 33 (2004), 2059–2074.MathSciNetCrossRefGoogle Scholar
  11. 11.
    S. R. Jammalamadaka and A. SenGupta, Topics in circular statistics, World Scientific, Singapore, 2001.CrossRefGoogle Scholar
  12. 12.
    S. Kato and M. C. Jones, A family of distributions on the circle with links to, and applications arising from, Möbius transformation, Amer. Statist. Assoc., 105 (2010), 249–262.MathSciNetCrossRefGoogle Scholar
  13. 13.
    S. Kim and A. SenGupta, A three parameter generalized von Mises distribution, Statist. Papers, 54 (2013), 685–693.MathSciNetCrossRefGoogle Scholar
  14. 14.
    K. V. Mardia, Statistics of directional data, Academic Press, New York, 1972.zbMATHGoogle Scholar
  15. 15.
    K. V. Mardia and P. E. Jupp, Directional Statistics, John Wiley, Chichester, 1999.CrossRefGoogle Scholar
  16. 16.
    J. A. Mooney, P. J. Helms, and I. T. Jolliffe, Fitting mixtures of von mises distributions: A case study involving sudden infant death syndrome, Comput. Statist. Data Anal., 41 (2003), 505–513.MathSciNetCrossRefGoogle Scholar
  17. 17.
    V. Papakonstantinou, Beiträge zur zirkulären Statistik, Ph.D. thesis, University of Zurich, Switzerland, 1979.Google Scholar
  18. 18.
    A. Pewsey, The wrapped skew-normal distribution on the circle, Comm. Statist: Theory Methods, 29 (2000), 2459–2472.MathSciNetCrossRefGoogle Scholar
  19. 19.
    A. Pewsey, Testing circular symmetry, Canad. J. Statist., 30 (2002), 591–600.MathSciNetCrossRefGoogle Scholar
  20. 20.
    A. Pewsey, The wrapped stable family of distributions as a flexible model for circular data, Comput. Statist. Data Anal., 52 (2008), 1516–1523.MathSciNetCrossRefGoogle Scholar
  21. 21.
    A. Pewsey, M. Neuhäuser, and G. D. Ruxton, Circular statistics in R, Oxford University Press, Croydon, 2013.zbMATHGoogle Scholar
  22. 22.
    A. Pewsey, K. Shimizu, and R. de la Cruz, On an extension of the von Mises distribution due to Batschelet, J. Appl. Stat., 38 (2011), 1073–1085.MathSciNetCrossRefGoogle Scholar
  23. 23.
    H. Y. Siew, S. Kato, and K. Shimizu, Generalized t-distribution on the circle, Jpn. J. Appl. Stat., 37 (2008), 1–16.CrossRefGoogle Scholar
  24. 24.
    D. Umbach and S. R. Jammalamadaka, Building asymmetry into circular distributions, Statist. Probab. Lett., 79 (2009), 659–663.MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian National Science Academy 2019

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of IsfahanIsfahanIran

Personalised recommendations