Contamination-robust expected risk and tail risk

Let the experiment in Statistical experiments have nonempty parameter set $\mathrm{\Theta }$. Choose a sigma-algebra $\mathcal{T}$ on $\mathrm{\Theta }$ and take $(\mathrm{\Theta },\mathcal{T})$ as its measurable action space (Measurable spaces and measurable sets). Specialize the target-first loss of Loss functions and risk to $L:\mathrm{\Theta }\times \mathrm{\Theta }\to [0,+\mathrm{\infty }]$ by taking its action space to be $(\mathrm{\Theta },\mathcal{T})$. Fix an integer sample size $N\ge 1$ and a measurable rule $\widehat{\theta }:{\mathcal{X}}^N\to \mathrm{\Theta }$. Equip ${\mathcal{X}}^N$ with the finite product sigma-algebra (The product sigma-algebra and its finite iterates); the iid sample law is the finite product measure supplied by The product measure of two sigma-finite measure spaces and For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique. Write $X(x)=x$ for the canonical sample vector on this product space. The Huber sampling law is from Huber contamination model, and nonnegative expectations use Expectation of a nonnegative or integrable random variable. For $0\le ϵ\le 1$ and $0\le \eta \le 1$, its Huber robust expected risk at fraction $ϵ$ is

$$\underset{\theta \in \mathrm{\Theta }}{\sup }\underset{Q}{\sup }{\mathbb{E}}_{P_{ϵ,\theta ,Q}^{\otimes N}}L(\theta ,\widehat{\theta }(X))\mathrm{.}$$

For the replacement risks, additionally assume ${\mathrm{\Delta }}_{\mathcal{X}}=\{(x,x):x\in \mathcal{X}\}\in \mathcal{A}\otimes \mathcal{A}$, as required by Adversarial sample replacement. Its replacement robust expected risk at fraction $\eta$ is the supremum of $\mathbb{E}L(\theta ,\widehat{\theta }(Y))$ over $\theta$ and all couplings in Adversarial sample replacement. For $t\in [0,+\mathrm{\infty }]$, the Huber tail risk at threshold $t$ is ${\sup }_{\theta ,Q}{P}_{ϵ,\theta ,Q}^{\otimes N}\{x:L(\theta ,\widehat{\theta }(x))\ge t\}$; the replacement tail risk is the supremum of $\mathrm{Pr}\{L(\theta ,\widehat{\theta }(Y))\ge t\}$ over $\theta$ and the same replacement couplings. Expected risk and tail risk are different criteria. The gross-error path in Gross-error contamination path is a path of laws; it is not a sample-replacement mechanism.