Is it possible to do better than Cramér Rao lower bound for different estimation methods

I’m new to learning about the Cramér Rao lower bound. Does the calculation of the CRLB imply that only particular estimation algorithms can be used (e.g. estimating delay and Doppler from a complex ambiguity surface) which satisfy the CRLB inequality? And, if I have additional algorithms to refine those estimates and get higher precision ones, could I ever do better than the CRLB?

Or in other words, does the CRLB always hold for any possible algorithm I come up with to compute/estimate the quantity of interest?

1 Answer

Does the calculation of the CRLB imply that only particular estimation algorithms can be used (e.g. estimating delay and Doppler from a complex ambiguity surface) which satisfy the CRLB inequality?

All unbiased estimators satisfy the CRLB.

You don't, however, have to use an unbiased estimator. Biased estimators might be lower in variance than the CRLB would allow for a biased estimator.

The existence of the CRLB doesn't force you to use any specific algorithm.

• Thanks for your insight @Marcus. The question I now have is how do I know if I am working with a biased estimator vs an unbiased one? To be more concrete, suppose I’m doing a MATLAB simulation (which I am) and I construct a complex ambiguity function surface to estimate my delay/Doppler. Then from this I refine my answers with a peak finding algorithm to better nail down my Doppler estimate to sub-Doppler bin accuracy. Lastly, I come up with another slick way to get an ever more refined estimate. In doing all of this is there an easy way to tell if this is an unbiased vs biased approach? – smigs Apr 21 '19 at 11:49
• well, do you have formula for the estimator you're using? – Marcus Müller Apr 21 '19 at 11:50
• It does! So, can you put that in a formula that begins with $\hat f_{\text{Doppler}} = \arg\max\limits_?\ldots$, replace the $?$ with your bin index symbol and add it to your question? – Marcus Müller Apr 21 '19 at 15:31
• Ok let me do that when I get home! Thanks for your help Marcus!!To keep it simple it’s just a 2D cross correlation formula between my received (noisy) signal and preamble acting as my template function (which is multiplying an exponential of different test Doppler frequencies). But it’s all being evaluated numerically. Then I look for the Doppler bin that contains the peak and that’s my initial Doppler estimate. I’m guessing that doesn’t help you out much. – smigs Apr 21 '19 at 15:33