Barankin Vector Locally Best Unbiased Estimates
This work addresses statistical estimation theory by providing a foundational extension for vector parameters, though it appears incremental as it builds on existing Barankin bound concepts.
The authors generalized the Barankin bound to vector cases in mean square error, deriving necessary and sufficient conditions to achieve this lower bound, which is expressed as a linear matrix inequality for unbiased estimator covariances.
The Barankin bound is generalized to the vector case in the mean square error sense. Necessary and sufficient conditions are obtained to achieve the lower bound. To obtain the result, a simple finite dimensional real vector valued generalization of the Riesz representation theorem for Hilbert spaces is given. The bound has the form of a linear matrix inequality where the covariances of any unbiased estimator, if these exist, are lower bounded by matrices depending only on the parametrized probability distributions.