Rethinking Biased Estimation: Improving Maximum Likelihood and the Cramér–Rao Bound
Foundations and Trends in Signal Processing
A lower bound on the Bayesian MSE based on the optimal bias function
IEEE Transactions on Information Theory
Optimality analysis of sensor-target localization geometries
Automatica (Journal of IFAC)
Adaptively biasing the weights of adaptive filters
IEEE Transactions on Signal Processing
Efficient recursive estimators for a linear, time-varying Gaussian model with general constraints
IEEE Transactions on Signal Processing
On parameter identifiability of MIMO radar with waveform diversity
Signal Processing
On the estimation of transfer functions, regularizations and Gaussian processes-Revisited
Automatica (Journal of IFAC)
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An important aspect of estimation theory is characterizing the best achievable performance in a given estimation problem, as well as determining estimators that achieve the optimal performance. The traditional Cramer-Rao type bounds provide benchmarks on the variance of any estimator of a deterministic parameter vector under suitable regularity conditions, while requiring a-priori specification of a desired bias gradient. 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), which is the sum of the variance and the squared-norm of the bias. Here, we develop bounds on the MSE in estimating a deterministic parameter vector x0 over all bias vectors that are linear in x0, which includes the traditional unbiased estimation as a special case. In some settings, it is possible to minimize the MSE over all linear bias vectors. More generally, direct minimization is not possible since the optimal solution depends on the unknown x0. Nonetheless, we show that in many cases, we can find bias vectors that result in an MSE bound that is smaller than the Cramer-Rao lower bound (CRLB) for all values of x0. Furthermore, we explicitly construct estimators that achieve these bounds in cases where an efficient estimator exists, by performing a simple linear transformation on the standard maximum likelihood (ML) estimator. This leads to estimators that result in a smaller MSE than the ML approach for all possible values of x0