A Volumetric Approach to Biased Estimation: Demonstration on Shrinkage Estimators

This work proposes a new approach, named as the volumetric design (VD), of developing biased estimators of deterministic parameters that are known in advance to belong to a compact subset in the parameter space. For analytical tractability, this approach is demonstrated on the choice of the shrinkage parameter of an estimator that scales the celebrated minimum variance unbiased estimator (MVUE) towards zero, where a spherical set is taken as the a priori knowledge on the parameters and the mean-squared error is adopted as the performance measure. Similar to the existing methods of the minimax (MX) and the deepest minimum criterion (DMC) estimators, the VD estimator also belongs to the class of admissible estimators that dominate the MVUE on the provided parameter (spherical) set. However, as its fundamental difference, it corresponds to the estimator that has the largest total relative volume on which it dominates the other estimators in this class, thereby achieving the best volumetric robustness in this manner.