Cramer-Rao bound for biased estimators

Question

So the Cramer-Rao bound gives us a lower bound on the variance of an estimator, now if the estimator is unbiased then we have a bound on the mean square error. While I can see the utility of the bound for unbiased estimators, i.e., we have a lower bound on the squared error from the parameter we are trying to estimate, how does the bound help us for biased estimators since if the estimator is biased then it's variance is not equal to its mean squared error, so we just have a bound seemingly unrelated to the parameter we are trying to estimate.

Answer 1

how does the bound help us for biased estimators since if the estimator is biased then it's variance is not equal to its mean squared error

If you are using the squared error loss then we may use the decomposition $$\text{Mean squared error} = \text{bias}^2 + \text{variance}$$

https://en.wikipedia.org/wiki/Mean_squared_error#Proof_of_variance_and_bias_relationship

Answer 2

The thing about unbiased estimators is that they are not always the best estimator in terms of minimized error. Often, you should choose to use a biased estimator. Nonetheless, one should still care about the variance of their estimator, as both bias and variance contribute to error: $$\mathbb E [(\hat \theta - \theta)^2]=b^2(\hat \theta) + \text{var}(\hat \theta)$$

Now say you have an estimator, and you are able to do the bias variance decomposition above in relation to some test-data. How might you asses whether your estimator is as precise as it can be given the bias? You can asses the variance with the CRB for biased estimators.

If the bias of your estimator is given by $b(\hat \theta)=\mathbb E[\hat \theta]-\theta$, then the CR bound is given by $$\text{var}(\hat \theta) \ge \frac{(1+b'(\hat \theta))^2}{I(\theta)}$$ where $b'(\hat \theta)$ is the derivative of the bias function. If there is no bias, i.e. $b(\hat \theta)=0$, then this reduces to the traditional bound: $$\text{var}(\hat \theta) \ge \frac{1}{I(\theta)}.$$

Proof found here.