Assume we have a model $p_\theta(\mathbf x)$ parameterized by $\theta$. Using Cramer-Rao lower bound, we are able to obtain a lower bound on the variance of the unbiased estimator of $\theta$ as $\mathbb{Var}[\hat{\theta}]>\epsilon$.
Now, consider a specific model $p_c(\mathbf x)\sim\mathcal{N}(0,c)$ for a known $c$, and assume an estimator achieves the above Cramer-Rao lower bound for $c$, i.e., $\mathbb{Var}[\hat{c}]=\epsilon$. Assuming one is going to use the model $\mathcal{N}(0,\hat{c})$, what can be said about the worst-case situation for this model in terms of being close to $\mathcal{N}(0,c)$? In other word, can $\hat{c}$ be bounded via $c$ and $\epsilon$ (something like $\hat{c}<c+f(\epsilon)$)?