×

A possibly asymmetric multivariate generalization of the Möbius distribution for directional data. (English) Zbl 1305.62206

Summary: A family of possibly asymmetric distributions on the unit hyper-disc with center at the origin is proposed. The paper presents a non-trivial multivariate generalization of the Möbius distribution on the unit disc. The family is obtained by applying a conformal mapping to the spherically symmetric beta distribution. The density functions of the family are unimodal, monotonic or uniantimodal. The conditional distribution of direction cosine given the length is a \(t\)-distribution on the sphere. The conditional distribution of the length given the direction cosine has a simple closed form expression, though not of any standard known distribution. Modality, skewness and direction parameters are globally orthogonal in the sense that the Fisher information matrix is diagonal. The proposed model on the hyper-disc, introducing this probability distribution for the very first time, is applied to an emerging area of astrophysics for a dataset on gamma-ray bursts and to a challenging area of geoinformatics for a dataset on worldwide earthquakes with magnitude greater than or equal to \(7.0 \mathrm{M}_{\mathrm{W}}\).

MSC:

62H11 Directional data; spatial statistics
62E15 Exact distribution theory in statistics
85A35 Statistical astronomy
86A32 Geostatistics
86A17 Global dynamics, earthquake problems (MSC2010)

Software:

circular; CircStats
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] (Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1972), Dover: Dover New York) · Zbl 0543.33001
[2] Amari, S., (Differential-Geometrical Methods in Statistics. Differential-Geometrical Methods in Statistics, Lecture Notes in Statistics, vol. 28 (1985), Springer: Springer Berlin) · Zbl 0559.62001
[3] Dortet-Bernadet, J. L.; Wicker, N., Model-based clustering on the unit sphere with an illustration using gene expression profiles, Biostatistics, 9, 66-80 (2008) · Zbl 1274.62761
[4] Fang, K.-T.; Kotz, S.; Ng, K.-W., Symmetric Multivariate and Related Distributions (1990), Chapman and Hall: Chapman and Hall London
[5] Gradshteyn, I. S.; Ryzhik, I. M., Table of Integrals, Series, and Products (2007), Elsevier: Elsevier Amsterdam · Zbl 1208.65001
[6] Huckemann, S. F.; Kim, P. T.; Koo, J.-Y.; Munk, A., Möbius deconvolution on the hyperbolic plane with application to impedance density estimation, Ann. Statist., 38, 2465-2498 (2010) · Zbl 1203.62055
[7] Jammalamadaka, S. R.; SenGupta, A., Topics in Circular Statistics (2001), World Scientific: World Scientific New Jersey
[8] Johnson, N. L.; Kotz, S.; Balakrishnan, N., Continuous Univariate Distributions, Vol. 2 (1995), Wiley: Wiley New York · Zbl 0821.62001
[9] Jones, M. C., Marginal replacement in multivariate densities, with application to skewing spherically symmetric distributions, J. Multivariate Anal., 81, 85-99 (2002) · Zbl 0998.60016
[10] Jones, M. C., The Möbius distribution on the disc, Ann. Inst. Statist. Math., 56, 733-742 (2004) · Zbl 1078.62009
[11] Jones, M. C.; Pewsey, A., A family of symmetric distributions on the circle, J. Amer. Statist. Assoc., 100, 1422-1428 (2005) · Zbl 1117.62365
[12] Kato, S., A distribution for a pair of unit vectors generated by Brownian motion, Bernoulli, 15, 898-921 (2009) · Zbl 1201.62066
[13] Kumaraswamy, P., A generalized probability density function for double-bounded random processes, J. Hydrol., 46, 79-88 (1980)
[14] Mardia, K. V.; Jupp, P. E., Directional Statisitcs (2000), Wiley: Wiley Chichester
[15] McCullagh, P., Some statistical properties of a family of continuous univariate distributions, J. Amer. Statist. Assoc., 84, 125-129 (1989) · Zbl 0676.62015
[16] McDonald, J. B., Some generalized functions for the size distribution of income, Econometrica, 52, 647-663 (1984) · Zbl 0557.62098
[17] Minh, D.-L.; Farnum, N. R., Using bilinear transformations to induce probability distributions, Comm. Statist. Theory Methods, 32, 1-9 (2003) · Zbl 1025.62003
[18] Ratcliffe, J. G., Foundations of Hyperbolic Manifolds (2006), Springer: Springer New York · Zbl 1106.51009
[19] SenGupta, A., On the constructions of probability distributions for directional data, Bull. Calcutta Math. Soc., 96, 139-154 (2004) · Zbl 1052.62061
[20] SenGupta, A., Probability distributions for directional data-short course, (The 1st Conference on Applied Probability and Statistical Methods. The 1st Conference on Applied Probability and Statistical Methods, Lecture Note (2010), IMS: IMS Brazil)
[21] SenGupta, A.; Kim, S.; Arnold, B. C., Inverse circular-circular regression, J. Multivariate Anal., 119, 200-208 (2013) · Zbl 1277.62187
[22] SenGupta, A.; Kulkarni, H. V.; Hubale, U. D.; SenGupta, A.; Smith, R., Prediction intervals for environmental events based on Weibull distribution, Advances in Statistical Methods for Emerging Environmental Problems. Advances in Statistical Methods for Emerging Environmental Problems, Environ. Ecol. Stat. (2014), Special Issue, Published online: 09 May 2014
[23] SenGupta, A.; Ugwuowo, F. I., A classification method for directional data with application to the human skull, Comm. Statist. Theory Methods, 40, 457-466 (2011) · Zbl 1208.62104
[24] Seshadri, V., A family of distributions related to the McCullagh family, Statist. Probab. Lett., 12, 373-378 (1991) · Zbl 0747.60018
[25] Shieh, G. S.; Johnson, R. A., Inferences based on a bivariate distribution with von Mises marginals, Ann. Inst. Statist. Math., 57, 789-802 (2005) · Zbl 1092.62062
[26] Shieh, G. S.; Zheng, S.; Johnson, R. A.; Chang, Y.-F.; Shimizu, K.; Wang, C.-C.; Tang, S.-L., Modeling and comparing the organization of circular genomes, Bioinformatics, 27, 912-918 (2011)
[27] Shimizu, K.; Iida, K., Pearson type VII distributions on spheres, Comm. Statist. Theory Methods, 31, 513-526 (2002) · Zbl 1009.62518
[28] Terras, A., Harmonic Analysis on Symmetric Spaces and Applications I (1985), Springer: Springer New York · Zbl 0574.10029
[29] Watson, G. S., Statistics on Spheres (1983), Wiley: Wiley New York · Zbl 0646.62045
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.