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Difference between Cramer Rao bound and mean absolute error MAE? I cannot see the difference, or where to use one and not the other. I know that Cramer and MAE are used to measure the quality of an estimator, but in what are they different?

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The Cramer-Rao bound is not a metric for the quality of a certain estimator, but rather can be calculated for a certain estimation problem and gives you a theoretical lower bound for the variance of any unbiased estimator for that problem (under some assumptions). For an unbiased estimator, the variance is equal to the expectation of the magnitude squared error, which is of course not the same as the absolute error.

If you have some estimator that you implemented, one thing that you can do is to compute a mean squared error and compare it to the CRB.

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