Tail Risk Measures

Definition

A tail risk measure maps a cost random variable to a scalar that depends only on the upper $\beta$-tail of its distribution, providing a principled middle ground between the over-conservative worst-case paradigm and the over-optimistic risk-neutral (expectation) paradigm for robotic autonomy. The canonical taxonomy borrowed from financial mathematics is VaR ⊂ CVaR ⊂ EVaR: the quantile, its coherent tail-average upper bound, and the tightest convex (Chernoff-based) upper bound. CVaR and EVaR are coherent; VaR is not.

Key Equations

A measure $\rho$ is a $\beta$-tail risk measure if it acts only through the $\beta$-tail $X_{\beta }=F_X^{-1}(1-\beta +\beta U)$, $U\sim \mathrm{Uniform}[0,1]$, i.e. $\rho (X)=\rho (X^{\prime })$ whenever $X_{\beta }{=}_d{X}_{\beta }^{\prime }$. The three canonical members form an ordered chain:
${\mathrm{VaR}}_{\beta }(X)\le {\mathrm{CVaR}}_{\beta }(X)\le {\mathrm{EVaR}}_{\beta }(X),$
so any of them being $\le 0$ certifies the chance constraint $\mathbb{P}(X\le 0)\ge 1-\beta$ — gloss: $\beta$ is a single tunable conservatism knob ($\beta \to 1$ risk-neutral, $\beta \to 0$ risk-averse).

Source Support

• akella2024risk — defines the $\beta$-tail (Def. 1), the VaR/CVaR/EVaR taxonomy and ordering, and the coherence status of each, as the unifying object for risk-aware planning, control, and verification in robotics.

Related Topics

• value_at_risk — the base (quantile) member, non-coherent.
• conditional_value_at_risk — the coherent tail-average.
• entropic_value_at_risk — the tightest convex upper bound.
• coherent_risk_measures — the axioms separating CVaR/EVaR from VaR.

Open Questions

• Which member of the chain best trades conservatism against tractability for a free-flying inspection planner under non-Gaussian state-estimation uncertainty?