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B0960
Title: Symmetry for distributions on functional spaces Authors:  Alicia Nieto-Reyes - Universidad de Cantabria (Spain) [presenting]
Heather Battey - Imperial College London (United Kingdom)
Abstract: Distributions on function spaces are intangible objects for which no formally defined notions of symmetry currently exist. This is a serious hindrance to the methodological development of modern statistics, as the absence of a notion of centre of symmetry precludes a well conceived generalisation of the sample median and other sample quantiles to functional data sets, which are ever more prevalent in a society overwhelmingly rich in data. After exposing, through examples, difficulties associated with crude extensions of multivariate notions of symmetry to function spaces, we propose a notion of symmetry for distributions on a general functional metric space $(F,d)$. Our notion is based on halfspaces induced by the metric $d$, and engenders symmetries in other domains, as we expose through simulations. In demonstrating that the corresponding sample centre of symmetry satisfies a robustness property for which the median was originally conceived, we ratify our sample centre of symmetry as a generalization of the sample median on function space. Finally, an application to handwriting data underlines the practical importance of our work.