Set-valued conditional functionals of random sets
Tobias Fissler, Ilya Molchanov
TL;DR
This work develops a unified framework for conditional set valued gauges of random sets by applying law determined scalar gauges to the support function of random closed convex sets. The construction yields a conditional set valued gauge $\mathsf{G}(\boldsymbol{X}|\mathfrak{A})$, defined as the largest $\mathfrak{A}$-measurable random closed convex set whose support in every direction is bounded by the conditional gauge of the support function. The authors connect this with fundamental objects such as conditional generalized expectations, conditional quantiles, and depth trimmed regions, and extend the theory to random cones and their translations. The framework generalizes cone distribution functions and cone expectiles to arbitrary gauge functions, enabling conditional risk assessment and depth based analysis in multiasset and vector-valued settings with information encoded by $\mathfrak{A}$.
Abstract
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.