Sticky Central Limit Theorems at Singularities

• Activity Number: 605
• Type: Topic Contributed
• Date/Time: Thursday, August 4, 2011 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Ezra Miller*+
• Companies: Duke University
• Keywords: Frechet mean ; central limit theorem ; algebraic variety ; polyhedron ; tree ; shape space

Abstract

Certain singular spaces, including algebraic varieties such as shape spaces (which parametrize configurations of points up to certain groups of motions) and polyhedral complexes such as tree spaces (which parametrize metric phylogenetic trees on fixed sets of taxa), increasingly arise as sample spaces in modern statistics problems. Applications to areas such as biology, medicine, and image analysis require understanding the asymptotics of distributions on such spaces. In the surprisingly common circumstance when Fr\'echet (intrinsic) means of distributions on singular spaces lie at singular points, central limit theorems can exhibit non-classical ``sticky'' behavior: positive mass can be supported on thin subsets of the ambient space. This talk reports on current investigations on this phenomenon by a Working Group at the Statistical and Applied Mathematical Sciences Institute (SAMSI) program on Analysis of Object Data.