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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2012.12762 |
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| _version_ | 1866915420498296832 |
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| author | Schötz, Christof |
| author_facet | Schötz, Christof |
| contents | A Fréchet mean of a random variable $Y$ with values in a metric space $(\mathcal Q, d)$ is an element of the metric space that minimizes $q \mapsto \mathbb E[d(Y,q)^2]$. 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 $\mathbb E[d(Y, q)^α]$ for $α> 0$, the minimizers of $\mathbb E[H(d(Y, q))]$ for integrals $H$ of non-decreasing functions, and the minimizers of $\mathbb E[\mathfrak c(Y, q)]$ for a quite unrestricted class of cost functions $\mathfrak c$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_12762 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Strong Laws of Large Numbers for Generalizations of Fréchet Mean Sets Schötz, Christof Probability Statistics Theory A Fréchet mean of a random variable $Y$ with values in a metric space $(\mathcal Q, d)$ is an element of the metric space that minimizes $q \mapsto \mathbb E[d(Y,q)^2]$. 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 $\mathbb E[d(Y, q)^α]$ for $α> 0$, the minimizers of $\mathbb E[H(d(Y, q))]$ for integrals $H$ of non-decreasing functions, and the minimizers of $\mathbb E[\mathfrak c(Y, q)]$ for a quite unrestricted class of cost functions $\mathfrak c$. 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. |
| title | Strong Laws of Large Numbers for Generalizations of Fréchet Mean Sets |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2012.12762 |