Uniform information errors control approximate D-optimal design gaps

Let $\mathcal{X}$ be a finite candidate set and let $\mathrm{\Xi }$ be a nonempty compact metric space of approximate designs on $\mathcal{X}$. For each $x$, let $J_x$ and $J_{h,x}$ be exact and computed sensitivities and let $W_x$ be symmetric positive definite. Suppose uniformly over $x\in \mathcal{X}$ that $\mathrm{\parallel }{J}_x{\mathrm{\parallel }}_2\le K$, $\mathrm{\parallel }{J}_{h,x}-J_x{\mathrm{\parallel }}_2\le \eta$, and $\mathrm{\parallel }{W}_x{\mathrm{\parallel }}_2\le w$, where $K,\eta ,w\ge 0$. Put $\delta =w(2K\eta +{\eta }^2)$. Suppose the criteria $\mathrm{\Phi }(\xi )=\log \det M(\xi )$ and ${\mathrm{\Phi }}_h(\xi )=\log \det M_h(\xi )$ are continuous on $\mathrm{\Xi }$, and for some $\alpha >0$ every matrix on every segment between $M(\xi )$ and $M_h(\xi )$ satisfies $v^{T}Av\ge \alpha \mathrm{\parallel }v{\mathrm{\parallel }}_2^2$. If ${\xi }^{\ast }$ maximizes $\mathrm{\Phi }$ and $\widehat{\xi }$ is a $\tau$-approximate maximizer of ${\mathrm{\Phi }}_h$, meaning ${\mathrm{\Phi }}_h(\widehat{\xi })\ge {\max }_{\xi \in \mathrm{\Xi }}{\mathrm{\Phi }}_h(\xi )-\tau$ with $\tau \ge 0$, then $\mathrm{\Phi }({\xi }^{\ast })-\mathrm{\Phi }(\widehat{\xi })\le \tau +\frac{2p\delta }{\alpha }\mathrm{.}$