I have a question about the estimated performance measurement process using Cramer Rao Bound.
Since Cramer Rao Bound (CRB) is the lower bound of the variance of any unbiased estimator, I thought it was right to compare it with MSE (Mean Squared Error), However, I found many papers compared CRB and Root Mean Squared Error (RMSE), and simulation results are also reasonable. I would really appreciate it if you could let me know why many researchers are comparing CRB and RMSE.
Thank you.
It is true that the Cramer Rao bound (CRB) is the lower bound of the variance of any unbiased estimator, and that the Mean Squared Error (MSE) is a commonly used performance metric for comparing estimators in signal processing and estimation theory. However, in practice, it is not always possible to achieve the CRB, and the MSE may not be an appropriate metric for comparing estimators if they are biased. The root mean square error (RMSE) is a commonly used metric for comparing biased estimators, since it accounts for both the bias and variance of the estimator. By comparing the RMSE to the CRB, researchers can determine whether their estimators are near-optimal with respect to the lower bound, even if they are biased.
In some cases, the CRB may not be analytically tractable, while the RMSE can be easily calculated from simulation results. Therefore, researchers may choose to compare the RMSE to the CRB because it provides a more practical measure of estimator performance that can be directly compared to simulation results. The choice of performance metric depends on the specific application and properties of the estimators being compared. While the CRB and MSE may be appropriate for some situations, the RMSE may provide a more practical measure of estimator performance in cases where an unbiased estimator is not achievable or when the CRB is not analytically tractable.
thank you so much for your kind response. It helped me a lot. There is one thing that I haven't cleared up yet. If the RMSE obtained from Monte Carlo simulation is close to the CRB, how can we say that it is close to the optimal estimator? This feels like a departure from the definition of a CRB. Thank you so much again.
The Cramér-Rao lower bound (CRB) is a theoretical lower bound on the variance of any unbiased estimator of a parameter based on a given set of measurements. If an estimator achieves the bound, it is said to be efficient. Thus, if the RMSE (root-mean-square error) obtained from a Monte Carlo simulation is close to the CRB, it suggests that the estimator being used is close to optimal, but it does not necessarily mean that the estimator is efficient. Efficiency is achieved only if the RMSE actually equals the CRB, which is often not possible in practice. Getting close to the CRB is an indication that the estimator is performing well and is a reasonable choice given the data and assumptions being used.
You are correct that the definition of efficiency for an estimator requires the mean square error (MSE) of the estimator to achieve the Cramér-Rao lower bound (CRB), not just the root-mean-square error (RMSE). However, in practice the RMSE is often used as a measure of the estimator's performance because it is easier to interpret and compare across different estimation methods or scenarios. The RMSE is not exactly the same as the square root of the CRB, a small RMSE value indicates that the estimator is performing well and is likely to be close to efficient. Therefore, comparing the RMSE to the CRB can provide a useful indication of how well an estimator is performing relative to the theoretical lower bound, even if the RMSE is not exactly equal to the square root of the CRB.
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