Academic Editor: A. Thavaneswaran
Copyright © 2010 Aaron Abrams et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider a problem in parametric estimation: given 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 . 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 .
By contrast, we give an example showing that, it does not hold for a certain
distribution on an infinite tree.