×

On the power of axial tests of uniformity on spheres. (English) Zbl 1442.62113

In the setting of directional data, that is, data distributed on the unit hypersphere \[ \mathcal{S}^{p-1}=\{\mathbf{x}\in\mathbb{R}^p : \|\mathbf{x}\|^2=1\} \] in \(\mathbb{R}^p\), the authors consider the problem of testing the null hypothesis of uniformly distributed data in cases where the data are axial, i.e., the data come from a distribution that has equal density on antipodal points. Alternative hypotheses are related to a semiparametric family of densities specified through a function \(f\), a vector \(\boldsymbol{\theta}\in\mathcal{S}^{p-1}\), and a concentration parameter \(\kappa\) (where \(\kappa=0\) gives a uniform distribution, regardless of the choice of \(\boldsymbol{\theta}\) or \(f\)).
In this setting, the authors first consider the case where \(\boldsymbol{\theta}\) is known, for which they prove a local asymptotic normality result which shows that a given test based on the sample covariance matrix of the data is locally asymptotically most powerful. Then, in the case where \(\boldsymbol{\theta}\) is unknown, the authors investigate properties of test statistics including that proposed by C. Bingham [Ann. Stat. 2, 1201–1225 (1974; Zbl 0297.62010)] and ones that take advantage of the ‘single-spiked’ nature of the alternatives under consideration. Simulations are used throughout to verify and illustrate the results obtained.

MSC:

62H11 Directional data; spatial statistics
62E17 Approximations to statistical distributions (nonasymptotic)
62F05 Asymptotic properties of parametric tests
62E20 Asymptotic distribution theory in statistics
62R30 Statistics on manifolds

Citations:

Zbl 0297.62010

Software:

rotasym
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] Anderson, T. W. (2003)., An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley, New York. · Zbl 1039.62044
[2] Anderson, T. W. and Stephens, M. A. (1972). Tests for randomness of directions against equatorial and bimodal alternatives., Biometrika 43 613-621. · Zbl 0255.62019
[3] Bernoulli, D. (1735). Quelle est la cause physique de l’inclinaison des plans des orbites des planètes? In, Recueil des pièces qui ont remporté le prix de l’Académie Royale des Sciences de Paris 1734 93-122. Académie Royale des Sciences de Paris, Paris. Reprinted in Daniel Bernoulli, Werke, Vol. 3, 226-303, Birkhäuser, Basel (1982).
[4] Bijral, A. S., Breitenbach, M. and Grudic, G. (2007). Mixture of Watson distributions: a generative model for hyperspherical embeddings. In, Artificial Intelligence and Statistics 35-42.
[5] Bingham, C. (1974). An antipodally symmetric distribution on the sphere., Ann. Statist. 2 1201-1225. · Zbl 0297.62010
[6] Chikuse, Y. (2003)., Statistics on Special Manifolds. Lecture Notes in Statistics 174. Springer, New York. · Zbl 1026.62051
[7] Cuesta-Albertos, J. A., Cuevas, A. and Fraiman, R. (2009). On projection-based tests for directional and compositional data., Stat. Comput. 19 367-380.
[8] 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
[9] Davies, R. B. (1977). Hypothesis testing when a nuisance parameter is present only under the alternative., Biometrika 64 247-254. · Zbl 0362.62026
[10] Davies, R. B. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative., Biometrika 74 33-43. · Zbl 0612.62023
[11] Davies, R. B. (2002). Hypothesis testing when a nuisance parameter is present only under the alternative: Linear model case., Biometrika 89 484-489. · Zbl 1023.62017
[12] Dryden, I. L. (2005). Statistical analysis on high-dimensional spheres and shape spaces., Ann. Statist. 33 1643-1665. · Zbl 1078.62058
[13] Dürre, A., Tyler, D. E. and Vogel, D. (2016). On the eigenvalues of the spatial sign covariance matrix in more than two dimensions., Statist. Probab. Lett. 111 80-85. · Zbl 1341.62118
[14] Fisher, N. I., Lewis, T. and Embleton, B. J. (1987)., Statistical analysis of spherical data. Cambridge Univ. Press press, Cambridge. · Zbl 0651.62045
[15] García-Portugués, E., Navarro-Esteban, P. and Cuesta-Albertos, J. A. (2020). On a projection-based class of uniformity tests on the hypersphere., Submitted.
[16] García-Portugués, E., Paindaveine, D. and Verdebout, T. (2020). On optimal tests for rotational symmetry against new classes of hyperspherical distributions., J. Amer. Statist. Assoc. To appear.
[17] García-Portugués, E. and Verdebout, T. (2018). An overview of uniformity tests on the hypersphere., arXiv preprint arXiv:1804.00286.
[18] Hallin, M. and Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity., Ann. Statist. 34 2707-2756. · Zbl 1114.62066
[19] Jupp, P. (2001). Modifications of the Rayleigh and Bingham tests for uniformity of directions., J. Multivariate Anal. 77 1-20. · Zbl 1044.62058
[20] Jupp, P. E. (2008). Data-driven Sobolev tests of uniformity on compact Riemannian manifolds., Ann. Statist. 36 1246-1260. · Zbl 1360.62218
[21] Lacour, C. and Pham Ngoc, T. M. (2014). Goodness-of-fit test for noisy directional data., Bernoulli 20 2131-2168. · Zbl 1357.62192
[22] Ley, C. and Verdebout, T. (2017)., Modern Directional Statistics. CRC Press, Boca Raton. · Zbl 1448.62005
[23] Mardia, K. V. and Jupp, P. E. (2000)., Directional Statistics. John Wiley & Sons, Chichester. · Zbl 0935.62065
[24] Paindaveine, D., Remy, J. and Verdebout, T. (2020). Sign tests for weak principal directions., Bernoulli. To appear. · Zbl 1439.62075
[25] Paindaveine, D. and Verdebout, T. (2016). On high-dimensional sign tests., Bernoulli 22 1745-1769. · Zbl 1360.62225
[26] Rayleigh, L. (1919). On the problem of random vibrations and random flights in one, two and three dimensions., Phil. Mag. 37 321-346. · JFM 46.1203.03
[27] Sra, S. and Karp, D. (2013). The multivariate Watson distribution: Maximum-likelihood estimation and other aspects., J. Multivariate Anal. 114 256-269. · Zbl 1258.62062
[28] Tyler, D. E. (1987). Statistical analysis for the angular central Gaussian distribution on the sphere., Biometrika 74 579-589. · Zbl 0628.62054
[29] 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.