Strong laws of large numbers for generalizations of Fréchet mean sets. Issue 1 (2nd January 2022)

• Record Type: Journal Article
• Authors: Schötz, Christof
• Is Part Of: Statistics. Volume 56:Issue 1(2022)
• Page Start: 34
• Page End: 52
• Publication Date: 2022-01-02
• Subjects: Fréchet mean -- strong law of large numbers -- set convergence; Mathematical statistics -- Periodicals; 519.505
• DOI: 10.1080/02331888.2022.2032063
• Languages: English
• ISSNs: 0233-1888

Abstract

Abstract : A Fréchet mean of a random variable Y with values in a metric space ( Q, d ) is an element of the metric space that minimizes q ↦ E d ( Y, q ) 2 . 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 E d ( Y, q ) α for α > 0, the minimizers of E H ( d ( Y, q ) ) for integrals H of non-decreasing functions, and the minimizers of E c ( Y, q ) for a quite unrestricted class of cost functions c . 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.