Computing the Barankin bound, by solving an unconstrained quadratic optimization problem

Abstract

The Barankin (1949) bound is the greatest lower bound on the variance of any unbiased estimate for a nonrandom parameter. Computing this bound yields, as a byproduct, an unbiased estimator that is at least locally best in the following sense. The estimator formula contains a reference parameter, when the unknown parameter happens to be equal to the reference, the variance of the estimate achieves the Barankin bound. If the dependence of the Barankin estimate on the reference parameter vanishes, then the estimate is also uniformly minimum variance. We obtain a simple derivation of the Barankin bound as the solution of an unconstrained convex quadratic optimization problem. In contrast the standard form of the Barankin bound involves the maximization of a ratio of quadratic quantities. For the case of PET inversion and natural gamma ray spectrometry, the Barankin estimate is only locally minimum variance, but it can be a viable alternative to the maximum likelihood estimate.