MSE bounds dominating the cramer-RAO bound

Traditional Cramer-Rao type bounds provide benchmarks on the variance of any estimator of a deterministic parameter vector, while requiring a priori specification of a desired bias gradient. However, in applications, it is often not clear how to choose the required bias. A direct measure of the estimation error that takes both the variance and the bias into account is the mean-squared error (MSE). Here, we develop bounds on the MSE in estimating a deterministic vector x₀ using estimators with linear bias vectors, which includes the traditional unbiased estimation as a special case. We show that there often exists linear bias vectors that result in an MSB bound that dominates the CRLB, so that it is smaller than the CRLB for all x₀ . Furthermore, we explicitly construct estimators that achieve these bounds by linearly transforming the maximum-likelihood estimator