Scholarlink

<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\varvec{X}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">X</mi> </mrow> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\varvec{X}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">X</mi> </mrow> </math></EquationSource> </InlineEquation> 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.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Set-valued conditional functionals of random sets

  • 期刊论文

  • 发表日期：

    2026年4月4日

查看全文
  • Tobias Fissler,
  • Ilya Molchanov

摘要

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 \({\varvec{X}}\) X . 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 \({\varvec{X}}\) X 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.