Strong Consistency of Set-Valued Frechet Sample Mean in Metric Spaces

• Activity Number: 691
• Type: Contributed
• Date/Time: Thursday, August 8, 2013 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Cedric Ginestet*+
• Companies: Boston University
• Keywords: Frechet mean ; Barycenter ; Karcher mean ; Metric space ; Set-valued Analysis ; Laws of large numbers

Abstract

The Frechet mean generalizes the idea of averaging in spaces where pairwise addition is not well-defined. In general metric spaces, the Frechet sample mean is not a consistent estimator of the theoretical Frechet mean. For non-trivial examples, sequences of Frechet sample mean sets may fail to converge in a set-analytical sense. We show that a specific type of almost sure (a.s.) convergence for the Frechet sample mean introduced by Ziezold (1977) is equivalent to the Kuratowski outer limit of a sequence of Frechet sample mean sets. Equipped with this outer limit, we prove different laws of large numbers for random variables taking values in separable (pseudo-)metric space with a finite metric. In this setting, we describe strong laws of large numbers for both the restricted and the non-restricted Frechet sample means. In particular, we demonstrate that all subsequences of Frechet sample means converge to a subset of the theoretical mean.