zbMATH — the first resource for mathematics

Some properties of a Cauchy family on the sphere derived from the Möbius transformations. (English) Zbl 1455.60032
Summary: We present some properties of a Cauchy family of distributions on the sphere, which is a spherical extension of the wrapped Cauchy family on the circle. The spherical Cauchy family is closed under the Möbius transformations on the sphere and the parameter of the transformed family is expressed using extended Möbius transformations on the compactified Euclidean space. Stereographic projection transforms the spherical Cauchy family into a multivariate \(t\)-family with a certain degree of freedom on Euclidean space. The Möbius transformations and stereographic projection enable us to obtain some results related to the spherical Cauchy family such as an efficient algorithm for random variate generation, a simple form of pivotal statistic and straightforward calculation of probabilities of a region. A method of moments estimator and an asymptotically efficient estimator are expressed in closed form. Maximum likelihood estimation is also straightforward.

60E05 Probability distributions: general theory
60D05 Geometric probability and stochastic geometry
60E10 Characteristic functions; other transforms
62E15 Exact distribution theory in statistics
Full Text: DOI Euclid
[1] Ahlfors, L.V. (1978). Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, 3rd ed. New York: McGraw-Hill.
[2] Barndorff-Nielsen, O.E. and Cox, D.R. (1994). Inference and Asymptotics. Monographs on Statistics and Applied Probability 52. London: CRC Press.
[3] Bhattacharya, R. and Patrangenaru, V. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds. I. Ann. Statist. 31 1-29. · Zbl 1020.62026
[4] Cutting, C., Paindaveine, D. and Verdebout, T. (2017). Testing uniformity on high-dimensional spheres against monotone rotationally symmetric alternatives. Ann. Statist. 45 1024-1058. · Zbl 1368.62133
[5] Cutting, C., Paindaveine, D. and Verdebout, T. (2017). Tests of concentration for low-dimensional and high-dimensional directional data. In Big and Complex Data Analysis. Contrib. Stat. 209-227. Cham: Springer. · Zbl 1381.62088
[6] Downs, T.D. (2003). Spherical regression. Biometrika 90 655-668. · Zbl 1436.62194
[7] Downs, T.D. and Mardia, K.V. (2002). Circular regression. Biometrika 89 683-697. · Zbl 1037.62056
[8] Dunau, J.-L. and Sénateur, H. (1988). Une caracterisation du type de la loi de Cauchy-conforme sur \(\mathbf{R}^n \). Probab. Theory Related Fields 77 129-135. · Zbl 0662.62008
[9] Ferguson, T. (1978). Maximum likelihood estimates of the parameters of the Cauchy distribution for samples of size 3 and 4. J. Amer. Statist. Assoc. 73 211-213.
[10] Ferguson, T.S. (1996). A Course in Large Sample Theory. Texts in Statistical Science Series. London: CRC Press. · Zbl 0871.62002
[11] Gabrielsen, G. (1986). Global maxima of real-valued functions. J. Optim. Theory Appl. 50 257-266. · Zbl 0577.90068
[12] García-Portugués, E. (2013). Exact risk improvement of bandwidth selectors for kernel density estimation with directional data. Electron. J. Stat. 7 1655-1685. · Zbl 1327.62241
[13] Gradshteyn, I.S. and Ryzhik, I.M. (2007). Table of Integrals, Series, and Products, 7th ed. San Diego, CA: Academic Press. · Zbl 1208.65001
[14] Hendriks, H. and Landsman, Z. (1996). Asymptotic behavior of sample mean location for manifolds. Statist. Probab. Lett. 26 169-178. · Zbl 0848.62017
[15] Holzmann, H., Munk, A., Suster, M. and Zucchini, W. (2006). Hidden Markov models for circular and linear-circular time series. Environ. Ecol. Stat. 13 325-347.
[16] Huckemann, S.F., Kim, P.T., Koo, J.-Y. and Munk, A. (2010). Möbius deconvolution on the hyperbolic plane with application to impedance density estimation. Ann. Statist. 38 2465-2498. · Zbl 1203.62055
[17] Iwaniec, T. and Martin, G. (2001). Geometric Function Theory and Non-linear Analysis. Oxford Mathematical Monographs. New York: Oxford University Press. · Zbl 1045.30011
[18] Jones, M.C. (2004). The Möbius distribution on the disc. Ann. Inst. Statist. Math. 56 733-742. · Zbl 1078.62009
[19] Kato, S. (2010). A Markov process for circular data. J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 655-672. · Zbl 1411.62256
[20] Kato, S. and Jones, M.C. (2010). A family of distributions on the circle with links to, and applications arising from, Möbius transformation. J. Amer. Statist. Assoc. 105 249-262. · Zbl 1397.60035
[21] Kato, S. and McCullagh P. (2020). Supplement to “Some properties of a Cauchy family on the sphere derived from the Möbius transformations.” https://doi.org/10.3150/20-BEJ1222SUPP
[22] Kato, S. and Pewsey, A. (2015). A Möbius transformation-induced distribution on the torus. Biometrika 102 359-370. · Zbl 1452.62188
[23] Kato, S., Shimizu, K. and Shieh, G.S. (2008). A circular-circular regression model. Statist. Sinica 18 633-645. · Zbl 1135.62044
[24] Kent, J.T. (1982). The Fisher-Bingham distribution on the sphere. J. Roy. Statist. Soc. Ser. B 44 71-80. · Zbl 0485.62015
[25] Kent, J.T., Hussein, I. and Jah, M.K. (2016). Directional distributions in tracking of space debris. In Proceedings of the 19th International Conference on Information Fusion (FUSION) 2081-2086.
[26] Kent, J.T. and Tyler, D.E. (1988). Maximum likelihood estimation for the wrapped Cauchy distribution. J. Appl. Stat. 15 247-254.
[27] Leipnik, R.B. (1947). Distribution of the serial correlation coefficient in a circularly correlated universe. Ann. Math. Stat. 18 80-87. · Zbl 0032.29504
[28] Letac, G. (1986). Seul le groupe des similitudes-inversions préserve le type de la loi de Cauchy-conforme de \(\mathbf{R}^n\) pour \(n>1\). J. Funct. Anal. 68 43-54. · Zbl 0612.60019
[29] Ley, C. and Verdebout, T. (2017). Modern Directional Statistics. Chapman & Hall/CRC Interdisciplinary Statistics Series. Boca Raton, FL: CRC Press. · Zbl 1448.62005
[30] Ley, C. and Verdebout, T. (2017). Skew-rotationally-symmetric distributions and related efficient inferential procedures. J. Multivariate Anal. 159 67-81. · Zbl 1368.62134
[31] Mardia, K.V. (1975). Statistics of directional data. J. Roy. Statist. Soc. Ser. B 37 349-393. · Zbl 0314.62026
[32] Mardia, K.V. and Jupp, P.E. (2000). Directional Statistics. Wiley Series in Probability and Statistics. Chichester: Wiley. · Zbl 0935.62065
[33] McCullagh, P. (1989). Some statistical properties of a family of continuous univariate distributions. J. Amer. Statist. Assoc. 84 125-129. · Zbl 0676.62015
[34] McCullagh, P. (1992). Conditional inference and Cauchy models. Biometrika 79 247-259. · Zbl 0753.62002
[35] McCullagh, P. (1996). Möbius transformation and Cauchy parameter estimation. Ann. Statist. 24 787-808. · Zbl 0859.62007
[36] McLachlan, G.J. and Krishnan, T. (2008). The EM Algorithm and Extensions, 2nd ed. Wiley Series in Probability and Statistics. Hoboken, NJ: Wiley Interscience. · Zbl 1165.62019
[37] Paindaveine, D. and Verdebout, T. (2017). Inference on the mode of weak directional signals: A Le Cam perspective on hypothesis testing near singularities. Ann. Statist. 45 800-832. · Zbl 1371.62043
[38] Ratcliffe, J.G. (2019). Foundations of Hyperbolic Manifolds, 3rd ed. Graduate Texts in Mathematics 149. New York: Springer. · Zbl 1430.51002
[39] Rudin, W. (1987). Real and Complex Analysis, 3rd ed. New York: McGraw-Hill. · Zbl 0925.00005
[40] Seshadri, V. (1991). A family of distributions related to the McCullagh family. Statist. Probab. Lett. 12 373-378. · Zbl 0747.60018
[41] Uesu, K., Shimizu, K. and SenGupta, A. (2015). A possibly asymmetric multivariate generalization of the Möbius distribution for directional data. J. Multivariate Anal. 134 146-162. · Zbl 1305.62206
[42] Watson, G.
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.