Hostname: page-component-cb9f654ff-hn9fh Total loading time: 0 Render date: 2025-08-23T11:38:31.591Z Has data issue: false hasContentIssue false
  • English
  • Français

A TRACTABLE SYMMETRIC MULTIVARIATE LOGISTIC DISTRIBUTION WITH BOUNDED SCORE FUNCTION

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  15 August 2025

PAUL G. BLACKWELL Open the ORCID record for PAUL G. BLACKWELL [Opens in a new window]
PAUL G. BLACKWELL*
Affiliation:
School of Mathematical and Physical Sciences, https://ror.org/05krs5044University of Sheffield, Sheffield S3 7RH, UK
*
e-mail: p.blackwell@sheffield.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Existing multivariate versions of the logistic probability distribution generally lack some of the useful properties of the univariate logistic distribution, such as its bounded score function or the tractability of its density function, or lack the rotational symmetry necessary for many applications. This paper clarifies some of the properties of such distributions and proposes a multivariate distribution closely related to the univariate logistic that has a tractable density, including the necessary normalising constant, bounded score function and elliptical symmetry. Some properties of its marginal distributions are explored, particularly in the bivariate case.

Keywords

multivariate logistic distributionbivariate distributionelliptical symmetry

MSC classification

Primary: 60E05: Distributions

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 7
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

The author gratefully acknowledges the support of the Leverhulme Trust through Research Fellowship RF-2020-241.

References

Ali, M., Mikhail, N. and Haq, M., ‘Class of bivariate distributions including bivariate logistic’, J. Multivariate Anal. 8(3) (1978), 405–412.10.1016/0047-259X(78)90063-5CrossRefGoogle Scholar
Borchers, H. W., ‘pracma: Practical Numerical Math Functions’, R package version 2.4.4 (2023). https://CRAN.R-project.org/package=pracma.Google Scholar
Bouchard-Côté, A., Vollmer, S. J. and Doucet, A., ‘The bouncy particle sampler: a nonreversible rejection-free Markov Chain Monte Carlo method’, J. Amer. Statist. Assoc. 113(522) (2018), 855–867.10.1080/01621459.2017.1294075CrossRefGoogle Scholar
Gumbel, E., ‘Bivariate logistic distributions’, J. Amer. Statist. Assoc. 56(294) (1961), 335–349.10.1080/01621459.1961.10482117CrossRefGoogle Scholar
Jensen, D. R., ‘Multivariate distributions’, in: Encyclopedia of Statistical Sciences, Vol. 6 (eds. Kotz, S., Johnson, N. L. and Read, C. B.) (Wiley, Hoboken, NJ, 1985), 43–55.Google Scholar
Kelker, D., ‘Distribution theory of spherical distributions and a location-scale parameter generalization’, Sankhya A 32 (1970), 419–430.Google Scholar
Kotz, S., Kozubowski, T. and Podgorski, K., The Laplace Distribution and Generalizations (Springer, New York, 2001).10.1007/978-1-4612-0173-1CrossRefGoogle Scholar
Malik, H. and Abraham, B., ‘Multivariate logistic distributions’, Ann. Statist. 1(3) (1973), 588–590.10.1214/aos/1176342430CrossRefGoogle Scholar
McGraw, D. and Wagner, J., ‘Elliptically symmetric distributions’, IEEE Trans. Inform. Theory 14(1) (1968), 110–120.10.1109/TIT.1968.1054081CrossRefGoogle Scholar
Michelot, T., Blackwell, P. G., Chamaillé-Jammes, S. and Matthiopoulos, J., ‘Inference in MCMC step selection models’, Biometrics 76(2) (2020), 438–447.10.1111/biom.13170CrossRefGoogle ScholarPubMed
Michelot, T., Blackwell, P. G. and Matthiopoulos, J., ‘Linking resource selection and step selection models for habitat preferences in animals’, Ecology 100 (2019), Article no. e02452.10.1002/ecy.2452CrossRefGoogle ScholarPubMed
Michelot, T., Gloaguen, P., Blackwell, P. G. and Etienne, M.-P., ‘The Langevin diffusion as a continuous-time model of animal movement and habitat selection’, Methods Ecol. Evol. 10(11) (2019), 1894–1907.10.1111/2041-210X.13275CrossRefGoogle Scholar
NIST, NIST Digital Library of Mathematical Functions (eds. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl and M. A. McClain), release 1.2.4 (2025). https://dlmf.nist.gov/.Google Scholar
R Core Team, R: A Language and Environment for Statistical Computing (R Foundation for Statistical Computing, Vienna, Austria, 2024). https://www.R-project.org/.Google Scholar
Volodin, N., ‘Spherically symmetric logistic distribution’, J. Multivariate Anal. 70(2) (1999), 202–206.10.1006/jmva.1999.1826CrossRefGoogle Scholar
Wang, Y. and Yin, C., ‘A new class of multivariate elliptically contoured distributions with inconsistency property’, Methodol. Comput. Appl. Probab. 23 (2021), 1377–1407.10.1007/s11009-020-09817-7CrossRefGoogle Scholar
Yin, C., Wang, Y. and Sha, X., ‘A new class of symmetric distributions including the elliptically symmetric logistic’, Comm. Statist. Theory Methods 51(13) (2022), 4537–4558.10.1080/03610926.2020.1818098CrossRefGoogle Scholar