# Cramer-Rao Lower Bound

In estimation problems, we may use Cramer-Rao Lower Bound (CRLB) to evaluate the best performance. But if there is no unbiased estimator can attain CRLB, what is the meaning of CRLB?

To clarify the question, I present a simple question here:

$$\alpha$$ is selected as the evaluating indicator of system $$A$$. We can obtain the CRLB of $$\alpha$$, but there is no estimation $$\hat \alpha$$ which can reach CRLB. Is it reasonable to use the CRLB of $$\alpha$$ to evaluate the performance of system $$A$$.

• biased estimators can have lower estimation error than an unbiased one. The CRLB is typically over optimistic in the low SNR region and doesn’t predict threshold performance either. It can only be trusted in the high SNR region . The Ziv-Zakai bounds are much better predictors but harder to calculate. CRLB isn’t the only tool in the box. The CRLB is relatively easier to calculate so it has a lot of utility. The fact that there are bounds on theoretical performance is often a surprise to many people who kludge algorithms. Brankin bounds like theZiv-Zakai don’t assume unbiasedness. – user28715 Mar 25 '19 at 16:48
• And what is the existence condition of Ziv-Zakai bounds? If you have any suggested literature... And if there is no estimator of $\alpha$ attaining the bound, how can we evaluate the performance of system $A$? For example, we can assume the metric of evaluation is the mean square error (MSE) of $\alpha$ and $\hat \alpha$. – Land Mar 25 '19 at 18:13
• if you look a google scholar, there are many references given but unfortunately, I don’t see any tutorials on the bound – user28715 Mar 25 '19 at 22:15
• Found this on Ziv–Zakai bound: Dinesh Ramasamy, Ziv-Zakai Bound for parameter estimation. They say: "Unlike the Cramér Rao Lower Bound, for which we only need the measurement model p(y|θ), we also need the prior p(θ) on the parameter θ in order to evaluate the ZZB." – Olli Niemitalo Mar 28 '19 at 5:25 