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July, 1973 On a Theorem of Bahadur on the Rate of Convergence of Point Estimators
James C. Fu
Ann. Statist. 1(4): 745-749 (July, 1973). DOI: 10.1214/aos/1176342469

Abstract

In this paper, we have proved a fundamental property of the characteristic function for the random variable $(\partial/\partial\theta) \log f(x \mid \theta)$. Based on this result, we have proved under regularity conditions different from Bahadur's that certain classes of consistent estimators $\{\theta_n^\ast\}$ are asymptotically efficient in Bahadur's sense $\lim_{\varepsilon \rightarrow 0} \lim_{n \rightarrow \infty} \frac{1}{n\varepsilon^2} \log P\theta\{|\theta_n^\ast - \theta| \geqq \varepsilon\} = -\frac{I(\theta)}{2}.$ Our proof also gives a simple and direct method to verify Bahadur's [2] result.

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James C. Fu. "On a Theorem of Bahadur on the Rate of Convergence of Point Estimators." Ann. Statist. 1 (4) 745 - 749, July, 1973. https://doi.org/10.1214/aos/1176342469

Information

Published: July, 1973
First available in Project Euclid: 12 April 2007

zbMATH: 0261.62028
MathSciNet: MR336893
Digital Object Identifier: 10.1214/aos/1176342469

Rights: Copyright © 1973 Institute of Mathematical Statistics

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Vol.1 • No. 4 • July, 1973
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