A Cramer-Rao Type Lower Bound for Estimators Satisfying a Bias Constraint

In this paper we give a Cramer-Rao (CR) type lower bound on estimator covariance which applies to any estimator whose bias gradient lies within a user specified ellipsoidal region of parameter space. In addition to providing a useful lower bound which is insensitive to small unknown estimator biases, the rate of change of the new bound provides a quantitative bias "sensitivity index" for the conventional bias-dependent CR bound. We give an analytical form for this sensitivity index which indicates that small estimator biases can make the new bound significantly less than the unbiased version of the CR bound when there exist important but difficult-to-estimate nuisance parameters. This implies that the application of the CR bound is unreliable for this situation due to severe bias sensitivity. As a practical illustration of these results, we consider the problem of estimating elements of the 2 x 2 covariance matrix associated with a pair of independent, identically distributed, zero-mean Gaussian random sequences.