Spherical Logistic Distribution

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.

Keywords

Bessel function / Logistic density / Bimodality / Spherical distribution / Rotational symmetry

References

[1] Abe T, Pewsey A. Sine-skewed circular distributions. Stat. Papers. 2011, 52 3 683-707

[2] Abramowitz M, Stegun IA. Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. 1965 New York: Dover Publications

[3] Azzalini A. A Class of Distributions which Includes the Normal Ones. Scand. J. Stat.. 1985, 12 2 171-178

[4] Banerjee A, Dhillon IS, Ghosh J, Sra S. Clustering on the unit hypersphere using von Mises-Fisher distributions. J. Mach. Learn. Res.. 2005, 6 1345-1382

[5] Bemmaor AC, Glady N. Modeling purchasing behavior with sudden death: a flexible customer lifetime model. Manag. Sci.. 2012, 58 5 1012-1021

[6] Everitt, B.S., Hothorn, T.: HSAUR2: A Handbook of Statistical Analyses Using R (2nd Edition). R package version 11-6. 2013; Available from: http://CRAN.R-project.org/package=HSAUR2

[7] Gatto R, Jammalamadaka SR. The generalized von Mises distribution. Stat. Methodol.. 2007, 4 3 341-353

[8] Hornik K, Grün B. movMF: An R package for fitting mixtures of von Mises-Fisher distributions. J. Stat. Softw.. 2014, 58 10 1-31

[9] Jupp PE, Kent JT. Fitting smooth paths to speherical data. Appl. Stat.. 1987, 36 1 34-46

[10] Kato S, Jones M . An extended family of circular distributions related to wrapped Cauchy distributions via Brownian motion. Bernoulli. 2013, 19 1 154-171

[11] Kent JT. The Fisher-Bingham distribution on the sphere. J. R. Stat. Soc. Ser. B (Methodological). 1982, 44 1 71-80

[12] Lewin L. Polylogarithms and Associated Functions. 1981 North Holland: Elsevier

[13] Mardia KV, Jupp PE. Directional Statistics. 2000 London: Wiley

[14] Meeker WQ, Escobar LA. Statistical Methods for Reliability Data. 1998 New York: Wiley

[15] Rosenthal M, Wu W, Klassen E, Srivastava A. Spherical regression models using projective linear transformations. J. Am. Stat. Assoc.. 2014, 109 508 1615-1624

[16] Ulrich G. Computer generation of distributions on the m-sphere. Appl. Stat.. 1984, 33 2 158-163

[17] Wood AT. Simulation of the von Mises Fisher distribution. Commun. Stat.-Simul. Comput.. 1994, 23 1 157-164

Funding

Iran National Science Foundation(No. 95014574)