zbMATH — the first resource for mathematics

Spherical logistic distribution. (English) Zbl 1442.62115
The authors define a spherical logistic distribution on the unit sphere \(\mathbb{S}^{d-1}\) in \(d\)-dimensional Euclidean space, with density function \[ f(y)=C_d(b,\kappa)\frac{e^{\kappa y^T\mu}}{(b-1+e^{\kappa y^T\mu})^2}\,, \] for \(y\in\mathbb{S}^{d-1}\), where \(\mu\in\mathbb{S}^{d-1}\), \(b\geq1\) and \(\kappa>0\) are parameters, and \(C_d(b,\kappa)\) is a normalizing constant. They consider method-of-moments and maximum likelihood estimation of these parameters, illustrating their results with simulations, with special attention given to the case \(d=3\). The paper concludes with an application of this spherical logistic distribution to a real data set.
62H11 Directional data; spatial statistics
62J12 Generalized linear models (logistic models)
62F10 Point estimation
62H12 Estimation in multivariate analysis
62R30 Statistics on manifolds
60E10 Characteristic functions; other transforms
62E15 Exact distribution theory in statistics
62H10 Multivariate distribution of statistics
60E05 Probability distributions: general theory
62P20 Applications of statistics to economics
Full Text: DOI
[1] Abe, T.; Pewsey, A., Sine-skewed circular distributions, Stat. Papers, 52, 3, 683-707 (2011) · Zbl 1434.62023
[2] Abramowitz, M.; Stegun, IA, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (1965), New York: Dover Publications, New York
[3] Azzalini, A., A Class of Distributions which Includes the Normal Ones, Scand. J. Stat., 12, 2, 171-178 (1985) · Zbl 0581.62014
[4] Banerjee, A.; Dhillon, IS; Ghosh, J.; Sra, S., Clustering on the unit hypersphere using von Mises-Fisher distributions, J. Mach. Learn. Res., 6, 1345-1382 (2005) · Zbl 1190.62116
[5] Bemmaor, AC; Glady, N., Modeling purchasing behavior with sudden death: a flexible customer lifetime model, Manag. Sci., 58, 5, 1012-1021 (2012)
[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., 4, 3, 341-353 (2007) · Zbl 1248.62012
[8] Hornik, K.; Grün, B., movMF: An R package for fitting mixtures of von Mises-Fisher distributions, J. Stat. Softw., 58, 10, 1-31 (2014)
[9] Jupp, PE; Kent, JT, Fitting smooth paths to speherical data, Appl. Stat., 36, 1, 34-46 (1987) · Zbl 0613.62086
[10] Kato, S.; Jones, M., An extended family of circular distributions related to wrapped Cauchy distributions via Brownian motion, Bernoulli, 19, 1, 154-171 (2013) · Zbl 1261.60019
[11] Kent, JT, The Fisher-Bingham distribution on the sphere, J. R. Stat. Soc. Ser. B (Methodological), 44, 1, 71-80 (1982) · Zbl 0485.62015
[12] Lewin, L., Polylogarithms and Associated Functions (1981), North Holland: Elsevier, North Holland · Zbl 0465.33001
[13] Mardia, KV; Jupp, PE, Directional Statistics (2000), London: Wiley, London
[14] Meeker, WQ; Escobar, LA, Statistical Methods for Reliability Data (1998), New York: Wiley, New York
[15] Rosenthal, M.; Wu, W.; Klassen, E.; Srivastava, A., Spherical regression models using projective linear transformations, J. Am. Stat. Assoc., 109, 508, 1615-1624 (2014) · Zbl 1368.62185
[16] Ulrich, G., Computer generation of distributions on the m-sphere, Appl. Stat., 33, 2, 158-163 (1984) · Zbl 0547.65095
[17] Wood, AT, Simulation of the von Mises Fisher distribution, Commun. Stat.-Simul. Comput., 23, 1, 157-164 (1994) · Zbl 0825.62022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.