Rethinking Biased Estimation: Improving Maximum Likelihood and the Cramér–Rao Bound
Foundations and Trends in Signal Processing
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We consider the problem of designing an estimation filter to recover a signal x[n] convolved with a linear time-invariant (LTI) filter h[n] and corrupted by additive noise. Our development treats the case in which the signal x[n] is deterministic and the case in which it is a stationary random process. Both formulations take advantage of some a priori knowledge on the class of underlying signals. In the deterministic setting, the signal is assumed to have bounded (weighted) energy; in the stochastic setting, the power spectra of the signal and noise are bounded at each frequency. The difficulty encountered in these estimation problems is that the mean-squared error (MSE) at the output of the estimation filter depends on the problem unknowns and therefore cannot be minimized. Beginning with the deterministic setting, we develop a minimax MSE estimation filter that minimizes the worst case point-wise MSE between the true signal x[n] and the estimated signal, over the class of bounded-norm inputs. We then establish that the MSE at the output of the minimax MSE filter is smaller than the MSE at the output of the conventional inverse filter, for all admissible signals. Next we treat the stochastic scenario, for which we propose a minimax regret estimation filter to deal with the power spectrum uncertainties. This filter is designed to minimize the worst case difference between the MSE in the presence of power spectrum uncertainties, and the MSE of the Wiener filter that knows the correct power spectra. The minimax regret filter takes the entire uncertainty interval into account, and as demonstrated through an example, can often lead to improved performance over traditional minimax MSE approaches for this problem