# 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
Commented Mar 25, 2019 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
Commented Mar 25, 2019 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
Commented Mar 25, 2019 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." Commented Mar 28, 2019 at 5:25

## 1 Answer

CRLB is usually used to assess the performance for an unbiased estimator because it is seen as the best variance you can achieve. If I understand your question correctly, you are asking if it is fair to use CRLB to assess the performance for a biased estimator? Well, this may not make much sense because you may come across situations where the biased estimator has better variance than the CRLB!

For example, Suppose the true value = 0, you can see one estimate is unbiased with large variance but the other has a bias with smaller variance.

• Thanks. I understand what you mean. I wonder what is the significance of CRLB, when I evaluate the performace of a estimator. I mean, the unbiased estimator with CRLB and biased estimator (such as small offset in your figure) with smaller variance, which is better? How to judge it generally?
– Land
Commented Mar 25, 2019 at 16:50
• Hi: the mse of an estimator is often used to combine the bias and the variance. I think expectation of mse = E(bias squared ) + the variance. ( But check me on that). So, it takes care of the case where the estimator is biased but has less variance. Commented Mar 26, 2019 at 5:38