 
Summary: OPTIMAL ESTIMATORS FOR THRESHOLDBASED
QUALITY MEASURES
AARON ABRAMS, SANDY GANZELL, HENRY LANDAU, ZEPH LANDAU, JAMES
POMMERSHEIM, AND ERIC ZASLOW
Abstract. We consider a problem in parametric estimation: given n
samples from an unknown distribution, we want to estimate which dis
tribution, from a given oneparameter family, produced the data. Fol
lowing Schulman and Vazirani [12], 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 translationinvariant, and
we conjecture that this conclusion also holds for any distribution on R.
By contrast, we give an example showing it does not hold for a certain
distribution on an infinite tree.
1. Introduction
Estimating probability distribution functions is a central problem in sta
tistics. Specifically, beginning with an unknown probability distribution on
an underlying space X, one wants to be able to do two things: first, given
some empirical data sampled from the unknown probability distribution,
