- Discussion paper:
- arXiv:2012.12762
- Title:
- Strong laws of large numbers for generalizations of Fréchet mean sets
- Author:
- Christof Schötz
- Abstract:
- A Fréchet mean of a random variable Y with values in a metric space (M,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)^a] for a>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.
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |
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