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 $A$.
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.