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Kurtosis test of modality for rotationally symmetric distributions on hyperspheres

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  • Kim, Byungwon
  • Schulz, Jörn
  • Jung, Sungkyu

Abstract

A test of modality of rotationally symmetric distributions on hyperspheres is proposed. The test is based on a modified multivariate kurtosis defined for directional data on Sd. We first reveal a relationship between the multivariate kurtosis and the types of modality for Euclidean data. In particular, the kurtosis of a rotationally symmetric distribution with decreasing sectional density is greater than the kurtosis of the uniform distribution, while the kurtosis of that with increasing sectional density is less. For directional data, we show an asymptotic normality of the modified spherical kurtosis, based on which a large-sample test is proposed. The proposed test of modality is applied to the problem of preventing overfitting in non-geodesic dimension reduction of directional data. The proposed test is superior than existing options in terms of computation times, accuracy and preventing overfitting. This is highlighted by a simulation study and two real data examples.

Suggested Citation

  • Kim, Byungwon & Schulz, Jörn & Jung, Sungkyu, 2020. "Kurtosis test of modality for rotationally symmetric distributions on hyperspheres," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
    • Handle: RePEc:eee:jmvana:v:178:y:2020:i:c:s0047259x19306141
    DOI: 10.1016/j.jmva.2020.104603
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    References listed on IDEAS

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    1. Eduardo García-Portugués & Davy Paindaveine & Thomas Verdebout, 2020. "On Optimal Tests for Rotational Symmetry Against New Classes of Hyperspherical Distributions," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(532), pages 1873-1887, December.
    2. Kollo, Tõnu, 2008. "Multivariate skewness and kurtosis measures with an application in ICA," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2328-2338, November.
    3. Byungwon Kim & Stephan Huckemann & Jörn Schulz & Sungkyu Jung, 2019. "Small‐sphere distributions for directional data with application to medical imaging," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 46(4), pages 1047-1071, December.
    4. Eduardo Garcia-Portugues & Davy Paindaveine & Thomas Verdebout, 2020. "On optimal tests for rotational symmetry against new classes of hyperspherical distributions," Post-Print hal-03169388, HAL.
    5. J. Koziol, 1987. "An alternative formulation of Neyman’s smooth goodness of fit tests under composite alternatives," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 34(1), pages 17-24, December.
    6. Sungkyu Jung & Ian L. Dryden & J. S. Marron, 2012. "Analysis of principal nested spheres," Biometrika, Biometrika Trust, vol. 99(3), pages 551-568.
    7. Loperfido, Nicola, 2020. "Some remarks on Koziol’s kurtosis," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
    8. Alsmeyer, Gerold, 1996. "Nonnegativity of odd functional moments of positive random variables with decreasing density," Statistics & Probability Letters, Elsevier, vol. 26(1), pages 75-82, January.
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    1. Arthur Pewsey & Eduardo García-Portugués, 2021. "Rejoinder on: Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 76-82, March.

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