Optimal Estimators for Threshold-Based Quality Measures

Abstract

We consider a problem in parametric estimation: given n samples from an unknown distribution, we want to estimate which distribution, from a given one-parameter family, produced the data. Following Schulman and Vazirani (2005), we evaluate an estimator in terms of the chance of being within a specified tolerance of the correct answer, in the worst case. We provide optimal estimators for several families of distributions on ℝ. We prove that for distributions on a compact space, there is always an optimal estimator that is translation invariant, and we conjecture that this conclusion also holds for any distribution on ℝ. By contrast, we give an example showing that, it does not hold for a certain distribution on an infinite tree.