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On trigonometric moments of the stereographic semicircular gamma distribution. (English) Zbl 1388.60050

Summary: Y. Phani [On stereographic circular and semicircular models. Namburu: Acharya Nagarjuna University (PhD Thesis) (2013)] constructed a good number of circular and semicircular models induced by inverse stereographic projection. D. Le Minh and N. R. Farnum [Commun. Stat., Theory Methods 32, No. 1, 1–9 (2003; Zbl 1025.62003)] and T. Abe et al. [“Symmetric unimodal models for directional data motivated by inverse stereographic projection”, J. Japan Statist. Soc. 40 No. 1, 45–61 (2010)] proposed a new method to derive circular distributions from the existing linear models. In this paper, a new semicircular model, which is coined as Stereographic Semicircular Gamma distribution is derived by inducing modified inverse stereographic projection on Gamma distribution. This distribution generalizes Stereographic Semicircular Exponential model [P. Yedlapalli et al., J. Appl. Probab. Stat. 8, No. 1, 75–90 (2013; Zbl 1307.62145)] and the density and distribution functions of proposed model admit closed form. Explicit expressions for trigonometric moments are derived by applying Meijer’s \(G\)-function and the new semicircular model is extended to construct Stereographic \(l\)-axial Gamma distribution.

MSC:

60E05 Probability distributions: general theory
62H11 Directional data; spatial statistics
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