Sticky central limit theorems on open books

Abstract

Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Fr\'{e}chet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension $1$ and hence measure $0$) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine. We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky).Comment: Published in at http://dx.doi.org/10.1214/12-AAP899 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Overview of object oriented data analysis

“…That area was put into a concise and elegant mathematical framework by Billera et al (2001), who developed a manifold stratified space for these data objects. This space has exhibited some perhaps surprising mathematical statistical behavior, in the concept of stickiness, as discovered by Hotz et al (2012). Stickiness is where the Fréchet mean of a data set (or probability distribution) remains totally fixed even when the data set (or distribution) is perturbed over a neighborhood, which is quite different from the behavior of the mean in other situations, for example in Euclidean robust statistics the mean has been criticized for gross instability even over very small neighborhoods.…”

“…While some classical theory has been developed, particularly on the circle and sphere, there do not appear to be any results of random matrix type, and only the preliminary paper of Sen et al (2008) in the HDLSS domain on a manifold. In the case of manifold stratified spaces, Hotz et al (2012) seems to be the only work to date.…”

Shape and object data analysis

“…Alternatively, one can work in the quotient space (Kendall, 1984), where one optimizes over registrations. The resulting quotient space representations of the data are usually on non-Euclidean spaces, where the geometry can be very complicated (Kendall et al, 1999), and include spaces with a manifold stratification (Hotz et al, 2013). Despite their different formulations, the ambient and quotient approaches to inference often lead to similar results in practice due to a Laplace approximation of the integral (Kenobi and Dryden, 2012).…”

Geometry and Statistics: Manifolds and Stratified Spaces

“…Approaches to stratified data spaces include the thesis of Bendich [25], who addressed the inverse problem of estimating stratified data spaces from data using persistent homology. Recently, the 2010-11 SAMSI working group Data Analysis on Sample Spaces with a Manifold Stratification made significant contributions to developing a statistical theory for stratified spaces [26,27].…”