Journal of Probability and Statistics
Volume 2010 (2010), Article ID 752750, 15 pages
doi:10.1155/2010/752750
Research Article

Optimal Estimators for Threshold-Based Quality Measures

Aaron Abrams,1 Sandy Ganzell,2 Henry Landau,3 Zeph Landau,4 James Pommersheim,5 and Eric Zaslow6

1Department of Mathematics and Computer Science, Emory University, USA
2Department of Mathematics and Computer Science, St. Mary's College of Maryland, USA
3AT&T Research, USA
4Department of Computer Science, University of California, Berkeley, USA
5James Pommersheim, Department of Mathematics, Reed College, USA
6Department of Mathematics, Northwestern University, USA

Received 26 February 2010; Revised 4 July 2010; Accepted 16 September 2010

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.

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.