Breaking the CRLB Barrier: Decreasing Mean Squared Error in Parameter Estimation Through Introduction of Regularization Bias

The inverse problem of parameter estimation arises throughout physics; one important example is the decaying biexponential (BE) or similar models describing phenomena as disparate as fluorescence decay, supernova light curves, and magnetic resonance relaxometry (MRR). However, BE parameter estimation is highly ill-posed, with results unstable with respect to noise. Therefore, we sought to incorporate Tikhonov regularization into the estimation problem to decrease variance at the expense of introducing a smaller amount of bias in order to decrease mean square error (MSE). In the difficult problem of estimating the four parameters of the BE model, we find that for typical signal-to-noise and reasonable MRR transverse relaxation (T2) ranges, the optimal regularization parameter can decrease MSE by an order of magnitude or more as compared to non-regularized least squares. While the mathematically optimal cannot be determined for experimental data with unknown generative parameters, standard approaches to selection of , such as generalized cross-validation, reliably yield improvements in MSE on the order of 50%. We also find that the variance of the estimators is below the Cramer-Rao lower bound, applicable only to unbiased estimators. In our primary application of MRR, these results will improve macromolecular mapping in the brain and other tissues.