Small sphere distributions and related topics in directional statistics

This dissertation consists of two related topics in the statistical analysis of directional data. The research conducted for the dissertation is motivated by advancing the statistical shape analysis to understand the variation of shape changes in 3D objects. The first part of the dissertation studies a parametric approach for multivariate directional data lying on a product of spheres. Two kinds of concentric unimodal-small subsphere distributions are introduced. The first kind coincides with a special case of the Fisher-Bingham distribution; the second is a novel adaption that independently models horizontal and vertical variations. In its multi-subsphere version, the second kind allows for correlation of horizontal variations over different subspheres. For both kinds, we provide new computationally feasible algorithms for simulation and estimation, and propose a large-sample test procedure for several sets of hypotheses. Working as models to fit the major modes of variation, the proposed distributions properly describe shape changes of skeletally-represented 3D objects due to rotation, twisting and bending. In particular, the multi-subsphere version of the second kind accounts for the underlying horizontal dependence appropriately. The second part is a proposal of hypothesis test that is applicable to the analysis of principal nested spheres (PNS). In PNS, determining which subsphere to fit, among the geodesic (great) subsphere and non-geodesic (small) subsphere, is an important issue and it is preferred to fit a great subsphere when there is no major direction of variation in the directional data. The proposed test utilizes the measure of multivariate kurtosis. The change of the multivariate kurtosis for rotationally symmetric distributions is investigated based on modality. The test statistic is developed by modifying the sample kurtosis. The asymptotic sampling distribution of the test statistic is also investigated. The proposed test is seen to work well in numerical studies with various data situations