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March, 1983 Convergence Rates of Estimators of a Finite Parameter: How Small can Error Probabilities Be?
Andrew L. Rukhin
Ann. Statist. 11(1): 202-207 (March, 1983). DOI: 10.1214/aos/1176346070


In this paper we investigate the rates of convergence to zero of error probabilities in the estimation problem of a finite-valued parameter. It is shown that if a consistent estimator attains Bahadur's bound for the probability of incorrect decision at some parametric point then the error probability does not tend to zero exponentially fast for some other value of the parameter. We evaluate the minimal possible rate of convergence of this probability at a fixed parametric point for all asymptotically minimax procedures, and establish a necessary and sufficient condition for any of these procedures to have a constant risk. A simple example is constructed to demonstrate the asymptotical inadmissibility of the usual maximum likelihood estimator.


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Andrew L. Rukhin. "Convergence Rates of Estimators of a Finite Parameter: How Small can Error Probabilities Be?." Ann. Statist. 11 (1) 202 - 207, March, 1983.


Published: March, 1983
First available in Project Euclid: 12 April 2007

zbMATH: 0506.62020
MathSciNet: MR684877
Digital Object Identifier: 10.1214/aos/1176346070

Primary: 62F12
Secondary: 62B10 , 62C20 , 62F10

Keywords: error probabilities , Finite-valued parameter , maximin procedures , weighted maximum likelihood estimator

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 1 • March, 1983
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