FIDUCIAL STATISTICS ON DIFFERENTIABLE MANIFOLDS

The primary aim of this thesis is to expand fiducial statistics in three directions. First, we develop new methods for generalized fiducial inference on the multivariate normal and binomial distributions. Our method using the \Cayley Transform to reparameterize the covariance of a multivariate normal distribution is of special note, since it leads to a posterior-like distribution on the covariance matrix that is not Wishart. Second, we introduce a general method to define Generalized Fiducial Distributions (GFDs) on constrained parameter spaces using principles from Riemannian Geometry. This method, called the Constrained GFD, takes a GFD from an ambient Euclidean space and redefines it onto a sub-Euclidean manifold. Third, we show that the concept of redefining a fiducial density onto a lower-dimensional measure-zero space generalizes greatly. Indeed, we define any GFD as the submanifold analogue of a higher-dimensional ambient density conditioned on the observed data. Parameter spaces suited for the Constrained GFD show up in a broad class of applications, including inference on Gaussian Graphical Models with fixed graph structures. As a review of the computational challenges to manifold learning, we consider a novel change-point problem from the Mayo Clinic in Rochester, MN.