Kurtosis test of modality for rotationally symmetric distributions on hyperspheres

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 S d . 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.

[1]  J. Koziol.  A Note on Measures of Multivariate Kurtosis , 1989 .

[2]  Martin Styner,et al.  Localized differences in caudate and hippocampal shape are associated with schizophrenia but not antipsychotic type , 2013, Psychiatry Research: Neuroimaging.

[3]  J. S. Marron,et al.  Principal Nested Spheres for Time-Warped Functional Data Analysis , 2013, 1304.6789.

[4]  Tamás F. Móri,et al.  On Multivariate Skewness and Kurtosis , 1994 .

[5]  D. Paindaveine,et al.  Inference on the mode of weak directional signals: a Le Cam perspective on hypothesis testing near singularities , 2015, 1512.04594.

[6]  Kanti V. Mardia,et al.  Torus principal component analysis with applications to RNA structure , 2018, The Annals of Applied Statistics.

[7]  J. Shohat,et al.  The problem of moments , 1943 .

[8]  On Kotz-Type Elliptical Distributions , 2005 .

[9]  T. Kollo.  Multivariate skewness and kurtosis measures with an application in ICA , 2008 .

[10]  J. S. Marron,et al.  Non-linear Hypothesis Testing of Geometric Object Properties of Shapes Applied to Hippocampi , 2015, Journal of Mathematical Imaging and Vision.

[11]  D. Paindaveine,et al.  Optimal Rank-Based Tests for the Location Parameter of a Rotationally Symmetric Distribution on the Hypersphere , 2015 .

[12]  Martin Styner,et al.  Non-Euclidean classification of medically imaged objects via s-reps , 2016, Medical Image Anal..

[13]  D. K. Hildebrand.  Kurtosis Measures Bimodality , 1971 .

[14]  J. S. Marron,et al.  Principal arc analysis on direct product manifolds , 2011, 1104.3472.

[15]  J. Koziol.  An alternative formulation of Neyman’s smooth goodness of fit tests under composite alternatives , 1987 .

[16]  S. Huckemann,et al.  Small‐sphere distributions for directional data with application to medical imaging , 2017, Scandinavian Journal of Statistics.

[17]  K. Mardia,et al.  Statistical Shape Analysis , 1998 .

[18]  P. O. Hulth,et al.  Search for a diffuse flux of astrophysical muon neutrinos with the IceCube 40-string detector , 2011, 1104.5187.

[19]  D. Paindaveine,et al.  On Optimal Tests for Rotational Symmetry Against New Classes of Hyperspherical Distributions , 2017, Journal of the American Statistical Association.

[20]  J. Marron,et al.  Analysis of principal nested spheres. , 2012, Biometrika.

[21]  Stephan F. Huckemann,et al.  Backward nested descriptors asymptotics with inference on stem cell differentiation , 2016, The Annals of Statistics.

[22]  Nicola Loperfido,et al.  Some remarks on Koziol's kurtosis , 2020, J. Multivar. Anal..

[23]  Allan Pinkus.  Totally Positive Matrices , 2009 .

[24]  Nonnegativity of odd functional moments of positive random variables with decreasing density , 1996 .

[25]  Martin Styner,et al.  TWO-STAGE EMPIRICAL LIKELIHOOD FOR LONGITUDINAL NEUROIMAGING DATA. , 2011, The annals of applied statistics.

[26]  K. Mardia,et al.  A small circle distribution on the sphere , 1978 .

[27]  N. Loperfido,et al.  Some inequalities between measures of multivariate kurtosis, with application to financial returns , 2012 .

[28]  J. S. Marron,et al.  Nested Sphere Statistics of Skeletal Models , 2013, Innovations for Shape Analysis, Models and Algorithms.

[29]  N. Loperfido.  A new kurtosis matrix, with statistical applications , 2017 .

[30]  J. S. Marron,et al.  Backwards Principal Component Analysis and Principal Nested Relations , 2014, Journal of Mathematical Imaging and Vision.

[31]  Distributions slanted to the right , 1995 .

[32]  K. Mardia.  Measures of multivariate skewness and kurtosis with applications , 1970 .

[33]  I. Dryden,et al.  Principal nested shape space analysis of molecular dynamics data , 2019, The Annals of Applied Statistics.

[34]  P. Westfall.  Kurtosis as Peakedness, 1905–2014. R.I.P. , 2014, The American statistician.

[35]  Stephan Huckemann,et al.  Dimension Reduction on Polyspheres with Application to Skeletal Representations , 2015, GSI.

[36]  Christophe Ley,et al.  Optimal R-estimation of a spherical location , 2011, 1109.4962.

[37]  Rabi Bhattacharya,et al.  Omnibus CLTs for Fr\'echet means and nonparametric inference on non-Euclidean spaces , 2013, 1306.5806.