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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.13620 |
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| _version_ | 1866913008444243968 |
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| author | Fissler, Tobias Molchanov, Ilya |
| author_facet | Fissler, Tobias Molchanov, Ilya |
| contents | Many key quantities in statistics and probability theory such as the expectation, quantiles, expectiles and many risk measures are law-determined maps from a space of random variables to the reals. We call such a law-determined map, which is normalised, positively homogeneous, monotone and translation equivariant, a gauge function. Considered as a functional on the space of distributions, we can apply such a gauge to the conditional distribution of a random variable. This results in conditional gauges, such as conditional quantiles or conditional expectations. In this paper, we apply such scalar gauges to the support function of a random closed convex set $\bX$. This leads to a set-valued extension of a gauge function. We also introduce a conditional variant whose values are themselves random closed convex sets. In special cases, this functional becomes the conditional set-valued quantile or the conditional set-valued expectation of a random set. In particular, in the unconditional setup, if $\bX$ is a random translation of a deterministic cone and the gauge is either a quantile or an expectile, we recover the cone distribution functions studied by Andreas Hamel and his co-authors. In the conditional setup, the conditional quantile of a random singleton yields the conditional version of the half-space depth-trimmed regions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13620 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Set-valued conditional functionals of random sets Fissler, Tobias Molchanov, Ilya Probability Statistics Theory 60D05 62H05 91G70 Many key quantities in statistics and probability theory such as the expectation, quantiles, expectiles and many risk measures are law-determined maps from a space of random variables to the reals. We call such a law-determined map, which is normalised, positively homogeneous, monotone and translation equivariant, a gauge function. Considered as a functional on the space of distributions, we can apply such a gauge to the conditional distribution of a random variable. This results in conditional gauges, such as conditional quantiles or conditional expectations. In this paper, we apply such scalar gauges to the support function of a random closed convex set $\bX$. This leads to a set-valued extension of a gauge function. We also introduce a conditional variant whose values are themselves random closed convex sets. In special cases, this functional becomes the conditional set-valued quantile or the conditional set-valued expectation of a random set. In particular, in the unconditional setup, if $\bX$ is a random translation of a deterministic cone and the gauge is either a quantile or an expectile, we recover the cone distribution functions studied by Andreas Hamel and his co-authors. In the conditional setup, the conditional quantile of a random singleton yields the conditional version of the half-space depth-trimmed regions. |
| title | Set-valued conditional functionals of random sets |
| topic | Probability Statistics Theory 60D05 62H05 91G70 |
| url | https://arxiv.org/abs/2504.13620 |