Two New Families of Local Asymptotically Minimax Lower Bounds in Parameter Estimation

TL;DR

This work develops two complementary families of local asymptotic minimax lower bounds for parameter estimation. The first family yields bounds under convex symmetric losses via a two-point MAP-error test, with Corollary 1 giving a local bound that sharpens constants by optimizing the prior weight; the second family derives bounds based on the minimum expected loss across multiple test points, with two- and three-point constructions that tighten constants beyond the standard two-point bounds. Collectively, the results provide computation-friendly, broadly applicable tools that require minimal regularity, extend to vector parameters, and exhibit correct decay rates in the number of observations, as demonstrated through diverse examples. The bounds offer practical guidance for benchmark-setting and performance characterization of estimators in settings where classical regularity conditions may fail or be overly conservative.

Abstract

We propose two families of asymptotically local minimax lower bounds on parameter estimation performance. The first family of bounds applies to any convex, symmetric loss function that depends solely on the difference between the estimate and the true underlying parameter value (i.e., the estimation error), whereas the second is more specifically oriented to the moments of the estimation error. The proposed bounds are relatively easy to calculate numerically (in the sense that their optimization is over relatively few auxiliary parameters), yet they turn out to be tighter (sometimes significantly so) than previously reported bounds that are associated with similar calculation efforts, across a variety of application examples. In addition to their relative simplicity, they also have the following advantages: (i) Essentially no regularity conditions are required regarding the parametric family of distributions; (ii) The bounds are local (in a sense to be specified); (iii) The bounds provide the correct order of decay as functions of the number of observations, at least in all examples examined; (iv) At least the first family of bounds extends straightforwardly to vector parameters.