Unknown operators, model mismatch, and robustness limits — Examples and Counterexamples

Let the truth space be $U=[1,2]\cup [4,5]$ with its absolute-value metric, the operator labels be $\mathfrak{A}=\{1,2\}$, and the data space be $\mathbb{R}$ with its absolute-value metric. For each $a\in \mathfrak{A}$, define $F_a:U\to \mathbb{R}$ by $F_a(x)=ax$. Let the truth-target map be $q_U:U\to Q:=U$, $q_U(x)=x$. Restrict the admissible truth/operator pairs to $\mathcal{C}=\{(x,1):1\le x\le 2\}\cup \{(x,2):4\le x\le 5\}\subseteq U\times \mathfrak{A},$ with the restriction to $\mathcal{C}$ of the metric $d_1((x,a),(x^{\mathrm{\prime }},a^{\mathrm{\prime }}))=\mathrm{\mid }x-x^{\mathrm{\prime }}\mathrm{\mid }+\mathrm{\mid }a-a^{\mathrm{\prime }}\mathrm{\mid }$ on ${\mathbb{R}}^2$. The first-coordinate projection ${\pi }_U:\mathcal{C}\to U$ is bijective, since every $x\in U$ occurs in exactly one admissible pair. Thus the configuration target $q_{\mathcal{C}}:=q_U\circ {\pi }_U:\mathcal{C}\to Q$, given by $q_{\mathcal{C}}(x,a)=x$, is also bijective and has equality as its recovery equivalence. For the stability-modulus definition, take $\mathcal{C}$ as the parameter class, with configuration forward map $G:\mathcal{C}\to \mathbb{R}$ given by $G(x,a)=F_a(x)=ax$ and target $q_{\mathcal{C}}$. Then $\omega (t)=t$ is a uniform stability modulus: for every $c,c^{\mathrm{\prime }}\in \mathcal{C}$, $\mathrm{\mid }{q}_{\mathcal{C}}(c)-q_{\mathcal{C}}(c^{\mathrm{\prime }})\mathrm{\mid }\le \omega (\mathrm{\mid }G(c)-G(c^{\mathrm{\prime }})\mathrm{\mid })$.

Let the truth space and target space both be $\mathbb{R}$ with their absolute-value metric, and take the action sigma-algebra to be the power set $\mathcal{P}(\mathbb{R})$, so every absolute metric-loss section is measurable. Let the admissible truth/operator pairs be $\mathbb{R}\times \{A_0,A_1\}$ and take target map $q(x)=x$. Define the linear forward operators $A_0(x)=x$ and $A_1(x)=x\mathrm{/}2$. Equip the data space $\mathbb{R}$ with its power-set sigma-algebra and usual addition, so every translation is measurable, and let $\nu$ be any probability measure on that measurable space, for example the Dirac measure at $0$. The specifications $(1,A_0,\nu )$ and $(2,A_1,\nu )$ have the same data law, since both forward signals equal $1$, while their targets are distance $1$ apart. Every randomized decision rule into the stated action space therefore has maximum expected absolute loss at least $1\mathrm{/}2$.