Open Access
October 2007 Optimal third root asymptotic bounds in the statistical estimation of thresholds
Franz Merkl, Leila Mohammadi
Ann. Statist. 35(5): 2193-2218 (October 2007). DOI: 10.1214/009053607000000325


This paper is concerned with estimating the intersection point of two densities, given a sample of both of the densities. This problem arises in classification theory. The main results provide lower bounds for the probability of the estimation errors to be large on a scale determined by the inverse cube root of the sample size. As corollaries, we obtain probabilistic bounds for the prediction error in a classification problem. The key to the proof is an entropy estimate. The lower bounds are based on bounds for general estimators, which are applicable in other contexts as well. Furthermore, we introduce a class of optimal estimators whose errors asymptotically meet the border permitted by the lower bounds.


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Franz Merkl. Leila Mohammadi. "Optimal third root asymptotic bounds in the statistical estimation of thresholds." Ann. Statist. 35 (5) 2193 - 2218, October 2007.


Published: October 2007
First available in Project Euclid: 7 November 2007

zbMATH: 1126.62022
MathSciNet: MR2363968
Digital Object Identifier: 10.1214/009053607000000325

Primary: 62G05
Secondary: 62G20

Keywords: Classification theory , entropy bounds , estimation error , nonparametric models , threshold-based classifiers

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 5 • October 2007
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