Abstract
Some of the spherical distributions can be constructed through proper transformation of the densities on plane. Since the logistic density on the Euclidean space has similar behavior to the normal distribution, it is of interest to extend it for spherical data. In this paper, we introduce spherical logistic distribution on the unit sphere and then study relevant statistical inferences including parameters estimation through method of moments and maximum likelihood techniques. It is shown that the spherical logistic distribution is a multimodal distribution with the marginal logistic density function. Proposed density has rotational symmetry property and this plays a key role to drive some important results related to first two moments. To investigate the proposed density in more details, some simulation studies along with analyzing real-life data are also considered.
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Acknowledgements
This research was in part supported by a Grant from Iran National Science Foundation [No. 95014574].
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Appendix
Appendix
To calculate the normalized constant of SL distribution represented in (2.1), we need to work with the spherical coordinate. To do so, we end up with
where
is the volume element of \({\mathbb {S}}^{d-2}\) for the spherical coordinate systems, identified by \(0\le \theta _j \le \pi \) for \(j=1,..,d-2\) and \(0\le \theta _{d-1} <2\pi .\) Note that the integral involved with all of \(\theta \)’s excepts \(\theta _1\) is equal to the volume of \({\mathbb {S}}^{d-2}, \) (see, for example, [13], for more details). Now, substituting \(\cos \theta _1\) with t we have
where \((1-t^2)^{(d-3)/2}=\sum _{i=0}^\infty (-1)^i {(d-3)/2\atopwithdelims ()i} t^{2i}\) and \({\text {Li}}_j\) is polylogarithmic function. This will complete our proof.
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Moghimbeygi, M., Golalizadeh, M. Spherical Logistic Distribution. Commun. Math. Stat. 8, 151–166 (2020). https://doi.org/10.1007/s40304-018-00171-2
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DOI: https://doi.org/10.1007/s40304-018-00171-2