.png){kind=icon}
.png){kind=icon}
Return
Set-valued conditional functionals of random sets
.png){kind=icon}
PRE
AI
DOI：10.1007/s00186-026-00923-9.png)
.png){kind=icon}
Original request for help.png){kind=icon}
Share.png){kind=icon}
Save
Abstract
En 中文
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 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 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.
Keywords:
Random set
Quantile
Random cone
Depth-trimmed region
Gaugefunction
Conditional distribution
Journal
M
IF:
1.2
Papers:
24
Citations:
0