UC Santa Barbara

This dissertation focuses mainly on directional data in two dimensions, called ``circular data," because such two-dimensional directions can be represented as points on the circumference of a unit circle. Such data, collected and analyzed by researchers in many scientific fields, needs special modeling and analysis. The thesis contains several somewhat independent results on the circular models and their analysis. First, a goodness-of-fit test for checking if a given dataset follows the wrapped stable distribution family is presented based on the empirical characteristic function. Then two dissimilarity measures for comparing any pair of curves around the circle are introduced and their use are explored in clustering such curves. This is followed by proving a result showing that wrapping a convolution of any number of linear components, yields the convolution of the corresponding wrapped distributions. Testing symmetry within the family of sine-skewed von Mises distributions is considered and compared with an existing test. The final result is a departure from the directional domain, and presents a Bayesian test for the number of modes in a two-component Gaussian mixture.