Statistical tools like the finite mixture models and model-based clustering have been used extensively in many fields such as natural language processing and genomic research to inves- tigate everything from copyright infringement to unraveling the mysteries of the evolutionary process. In model-based clustering, the samples are assumed to be realizations of a mixture distribution consisting of one or more mixture components, and the model attempts to discern what this original model is, given the observed data. In our investigation we explore directional distributions on the circle, the sphere, and the hypersphere, where the component distributions are themselves respectively the von Mises distributions in 2-dimensions, the von Mises-Fisher distributions in 3-dimensions, and 𝑝-dimensional von Mises-Fisher distributions for large 𝑝. In each case, the observations lie on the circle, the unit-sphere, or the hypersphere 𝕊𝑝−1 embedded in R𝑝, due to the inherent structure of the data, or by normalizing the curves. We look specifically at clustering curves around the unit circle 𝕊1, treating them first as mixture distributions, and in an alternate approach, as functional data that can be explored via their Fourier coefficients. We also investigate clustering high-dimensional, extremely sparse textual data, by looking at twitter data from the day of the 2016 United States presidential election as document vectors on the unit hypersphere. Finally, we introduce and discuss a broad family of spherical distributions that we call the “Generalized Fisher-Bingham family," and present details of a software package that we developed to simulate and visualize members of this family.

We study interval estimation for parameters of discrete distributions, focusing on the

binomial, Poisson, negative binomial, and hypergeometric distributions explicitly. We

provide a broad treatment of the problem, covering both conventional and randomized

confidence intervals, as well as Geyer and Meeden’s concept of fuzzy confidence intervals.

We take a graphical approach to the problem through the use of coverage probability

functions and determine the optimal procedure under each of a wide variety of criteria,

including multiple notions of length. Several new methods are proposed, including a

method that produces length optimal fuzzy confidence intervals. Credible intervals and

multi-parameter discrete distributions are also considered.

This thesis investigates two different problems relating to circular data. One relates to change-point problems. Tests in this context are meant to detect the point in time at which a sample of observations changes the probability distribution from which they came. Suppose one has a set of independent vectors of measurements, observed in a time-ordered or space-ordered sequence. In our set-up, these observations are circular data and we are interested as to which point in time does the distribution change from having one mode to having more than one mode. In this work we model unimodality or bimodality with a mixture of two Circular Normal distributions, which admits both possibilities, albeit for different parameter values. Tests for detecting the change-point are derived using the generalized likelihood ratio method. We obtain simulated distri- butions and critical values for the appropriate test statistics in finite samples, as well as provide the asymptotic distributions, under some regularity conditions. We also tackle this problem from a Bayesian perspective. In the second part, the goal is to estimate the concentration parameter of a Circular Normal distribution when the mean direction is unknown. We present two alternate approaches that incorporate prior knowledge on the mean direction (i) via a preliminary test on the mean direction, the so-called “preliminary test estimators” and (ii) through an assumed prior distribution on the mean direction as one does in Bayes procedures. We compare such alternate estimators with the standard maximum likelihood estimator and explore when one method is superior to the other.