Beran Estimation of the Fisher-Bingham and Curved Kent Distributions

Mudalige, Nishan
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University of Guelph

A directional measurement can be considered as a point incident to the unit hyper-sphere S^(p-1). Techniques for developing statistical tools to analyze directional data have a long and established history. Due to this interest in directional statistics, several probability distributions have been developed for data over S^(p-1). A general exponential model which embodies some classical distributions as special cases has been proposed in high dimensions and is the subject of this dissertation. We describe the relationship between this general exponential model and the generalized form of another popular distribution over S^(p-1) for all p >= 3. We extend our results by developing a framework for fitting several models over S^(p-1) by a regression estimator and utilizing machine learning techniques. We apply these models to fit astronomical data over S^2 and also on simulated data over higher dimensions. We also examine the goodness of fit and asymptotic properties of our models.

directional statistics, exponential family, maximum likelihood estimation, spherical harmonics, hyper-sphere, canonical exponential family, curved exponential family, traceless matrices