A Cramer-Rao Type Lower Bound for Essentially Unbiased Parameter Estimation

Abstract
In this report a new Cramer-Rao (CR) type lower bound is derived which takes into account a user-specified constraint on the length of the gradient of estimator bias with respect to the set of underlying parameters. If the parameter space is bounded, the constraint on bias gradient translates into a constraint on the magnitude of the bias itself; the bound reduces to the standard unbiased form of the CR bound for unbiased estimation. In addition to its usefulness as a lower bound that is insensitive to small biases in the estimator, the rate of change of the new bound provides a quantitative bias sensitivity index for the general bias-dependent CR bound. An analytical from for this sensitivity index is derived which indicates that small estimator biases can make the new bound significantly less than the unbiased CR bound when important but difficult-to-estimate nuisance parameters exist. 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, the problem of estimating elements of the 2 x 2 covariance matrix associated with a pair of independent identically distributed (IID) zero-mean Gaussian random sequences is presented.

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