In earlier studies of circular data, the corresponding probability distributions considered were mostly assumed to be symmetric. However, the assumption of symmetry may not be meaningful for some data. Thus there has been increased interest, more recently, in developing skewed circular distributions. In this article we introduce three skewed circular models based on inverse stereographic projection (ISP), originally introduced by Minh and Farnum (Comput. Stat.–Theory Methods, 32, 1–9, 2003), by considering three different versions of skewed-t distribution on real line considered in the literature, namely skewed-t by Azzalini (Scand. J. Stat., 12, 171–178, 1985), two-piece skewed-t, (seemingly first considered in Gibbons and Mylroie Appl. Phys. Lett., 22, 568–569, 1973 and later by Fernández and Steel J. Amer. Statist. Assoc., 93, 359–371 1998) and skewed-t by Jones and Faddy (J. R. Stat. Soc. Ser. B (Stat. Methodol.), 65, 159–174, 2003). Unimodality and skewness of the resulting distributions are addressed in this paper. Further, real data sets are used to illustrate the application of the new models. It is found that under certain condition on the original scaling parameter, the resulting distributions may be unimodal. Furthermore, the study in this paper concludes that ISP circular distributions obtained from skewed distributions on the real line may provide an attractive alternative to other asymmetric unimodal circular distributions, especially when combined with a mixture of uniform circular distribution.
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The partial support of this research through a Discovery Grant from NSERC, Canada to Yogendra Chaubey is gratefully acknowledged. The authors would also like to acknowledge the excellent suggestions by two reviewers, that have substantially improved the presentation and content of the paper. The authors would also like to thank Dr. Arthur Pewsey for directing us to his research web site for the data sets used in this paper.
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Part of this research is based on Shamal Karmaker’s M.Sc. thesis presented in the Department of Mathematics and Statistics at Concordia University.
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Chaubey, Y.P., Karmaker, S.C. On Some Circular Distributions Induced by Inverse Stereographic Projection. Sankhya B 83, 319–341 (2021). https://doi.org/10.1007/s13571-019-00201-1