Abstract/Details

Nonparametric statistics on manifolds with applications to shape spaces

Bhattacharya, Abhishek. 
 The University of Arizona ProQuest Dissertations Publishing,  2008. 3336565.

Abstract (summary)

This thesis presents certain recent methodologies and some new results for the statistical analysis of probability distributions on non-Euclidean manifolds. The notions of Fréchet mean and variation as measures of center and spread are introduced and their properties are discussed. The sample estimates from a random sample are shown to be consistent under fairly broad conditions. Depending on the choice of distance on the manifold, intrinsic and extrinsic statistical analyses are carried out. In both cases, sufficient conditions are derived for the uniqueness of the population means and for the asymptotic normality of the sample estimates. Analytic expressions for the parameters in the asymptotic distributions are derived. The manifolds of particular interest in this thesis are the shape spaces of k-ads. The statistical analysis tools developed on general manifolds are applied to the spaces of direct similarity shapes, planar shapes, reflection similarity shapes, affine shapes and projective shapes. Two-sample nonparametric tests are constructed to compare the mean shapes and variation in shapes for two random samples. The samples in consideration can be either independent of each other or be the outcome of a matched pair experiment. The testing procedures are based on the asymptotic distribution of the test statistics, or on nonparametric bootstrap methods suitably constructed. Real life examples are included to illustrate the theory.

Indexing (details)


Subject
Mathematics;
Statistics
Classification
0405: Mathematics
0463: Statistics
Identifier / keyword
Pure sciences; Extrinsic analysis; Frechet analysis; Intrinsic analysis; Manifolds; Nonparametric inference; Shape spaces; Shapes of k-ads
Title
Nonparametric statistics on manifolds with applications to shape spaces
Author
Bhattacharya, Abhishek
Number of pages
152
Degree date
2008
School code
0009
Source
DAI-B 69/11, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
978-0-549-90104-4
Advisor
Bhattacharya, Rabi
Committee member
Glickenstein, David; Kennedy, Thomas G.; Pickrell, Douglas M.; Shaked, Moshe
University/institution
The University of Arizona
Department
Mathematics
University location
United States -- Arizona
Degree
Ph.D.
Source type
Dissertation or Thesis
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
3336565
ProQuest document ID
304685224
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
https://www.proquest.com/docview/304685224/abstract