Strong laws of large numbers for generalizations of Fréchet mean sets

A Fréchet mean of a random variable Y with values in a metric space is an element of the metric space that minimizes . This minimizer may be non-unique. We study strong laws of large numbers for sets of generalized Fréchet means. Following generalizations are considered: the minimizers of for , the...

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Bibliographic Details
Published inStatistics (Berlin, DDR) Vol. 56; no. 1; pp. 34 - 52
Main Author Schötz, Christof
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 02.01.2022
Taylor & Francis Ltd
Subjects
Cost function
Fréchet mean
Metric space
Random variables
set convergence
strong law of large numbers
Yttrium
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Summary:A Fréchet mean of a random variable Y with values in a metric space is an element of the metric space that minimizes . This minimizer may be non-unique. We study strong laws of large numbers for sets of generalized Fréchet means. Following generalizations are considered: the minimizers of for , the minimizers of for integrals H of non-decreasing functions, and the minimizers of for a quite unrestricted class of cost functions . We show convergence of empirical versions of these sets in outer limit and in one-sided Hausdorff distance. The derived results require only minimal assumptions.
ISSN:0233-1888
1029-4910
DOI:10.1080/02331888.2022.2032063