Nonparametric bootstrap method on Stiefel manifold and GPAV algorithm for ASP fit

The first part of this thesis studies statistical models for data on a Stiefel Manifold. The latter is the collection of n by p matrices X that satisfy the restriction X' X = I. Such a matrix is commonly used to describe a rigid configuration of p distinguishable directions in n dimensional space. We begin by defining the intrinsic mean of a random element that has an unknown probability distribution on the Stiefel manifold. Given a random sample of observations from this distribution, we devise a consistent estimator, within the manifold, for the unknown intrinsic mean. A nonparametric bootstrap confidence region is then constructed for the intrinsic mean. The coverage probability of this confidence region is proven to converge to the desired level. As an application, the confidence region is used to analyze a set of vectorcardiogram data. The second part of this thesis addresses the lack of an efficient algorithm to realize the ASP method for low-risk estimation of mean responses, using bi-monotone shrinkage with respect to a well-chosen basis, in two-way layouts where both of the factors are ordinal. Our solution for small two-way layouts draws on a Generalized Pool Adjacent Violators algorithm, of relatively low computational complexity, that closely approximates the desired ASP estimator. The latter minimizes an estimated risk criterion over the bi-monotone shrinkage class. Further algorithmic refinements make feasible the closely approximate construction of low risk bi-monotone shrinkage ASP fits in larger two-way layouts.