# Worst-case error related to Cramer-Rao bound

I would like to understand the relation (if any) between the Cramer-Rao Lower Bound of estimation theory and the following simple definition of "reconstruction accuracy" which doesn't use any statistical framework.

Let $\theta\in\Theta\subset \mathbb{R}^n$ be an unknown parameter vector and let $x=f(\theta)\subset\mathbb{R}^m$ be a set of measurements, from which $\theta$ is to be reconstructed. Assume that the absolute error in each measurement is bounded from above by $\epsilon>0$, and otherwise nothing is known about the measurement error. Let $T_{\theta,\epsilon}=f^{-1}(x+B(0,\epsilon))$ (the full preimage, always well-defined). Obviously, $\theta\in T_{\theta,\epsilon}$. The "best possible" reconstruction accuracy is defined as $\delta_{\theta,\epsilon}=\sup_{t\in T_{\theta,\epsilon}} \|t-\theta\|$.

(Informal justification for the definition: if the measurement error is bounded by $\epsilon$, the original parameter vector $\theta$ can only be recovered up to accuracy $\delta_{\theta,\epsilon}$ because any parameter vector $\hat\theta\in T_{\theta,\epsilon}$ could have produced a measurement vector $\hat x \in B(x,\epsilon)$.)

In contrast, Cramer-Rao lower bound depends on some distribution $p(x;\theta)$ of the measurements, and gives a bound on the variance of any unbiased estimator $\hat \theta$ of $\theta$ satisfying some regularity conditions.

Apparently, the two notions are different and in general not comparable.

Still, are there cases in which the Cramer-Rao bound can be thought of as the $\delta_{\theta,\epsilon}$ defined above?

## Example

Consider the simple case of a single parameter. Assuming $f'(\theta)\neq 0$, first-order approximation of the reconstruction accuracy as defined above is $$\delta_{\theta,\epsilon}\approx \frac{\epsilon}{f'(\theta)}. \qquad (1)$$ On the other hand, assuming that the measurements are modeled as $$x = s(\theta) + w$$ where $w$ is Gaussian with zero mean and variance $\sigma^2$, we have (see e.g. Kay's book, section 3.5) $$CRB\{\theta\} = \frac{\sigma^2}{\biggl(\frac{\partial s}{\partial \theta}\biggr)^2} \qquad (2)$$

Comparing $(1)$ and $(2)$ seems to suggest that in this case the notions are very closely related.

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It's not immediately clear to me that any definition of "unbiased" can make sense in your problem involving "reconstruction accuracy". The Cramer–Rao bound does not apply to biased estimators: there are biased estimators whose mean squared error is less than the Cramer–Rao bound. – Michael Hardy May 23 '12 at 17:06
the modification of the Cramer-Rao bound for biased estimators is simple and standard. – mike May 23 '12 at 22:27
Added the "Example" in the main post. – dima_b May 24 '12 at 9:36