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June, 1983 On Moderate Deviation Theory in Estimation
Wilbert C. M. Kallenberg
Ann. Statist. 11(2): 498-504 (June, 1983). DOI: 10.1214/aos/1176346156


The performance of a sequence of estimators $\{T_n\}$ of $\theta$ can be measured by the probability concentration of the estimator in an $\varepsilon_n$-neighborhood of $\theta$. Classical choices of $\varepsilon_n$ are $\varepsilon_n = cn^{-1/2}$ (contiguous case) and $\varepsilon_n = \varepsilon$ fixed for all $n$ (non-local case). In this article all sequences $\{\varepsilon_n\}$ with $\lim_{n\rightarrow\infty} \varepsilon_n = 0$ and $\lim_{n\rightarrow\infty} \varepsilon_nn^{1/2} = \infty$ are considered. In that way the statistically important choices of small $\varepsilon$'s are investigated in a uniform sense; in that way the importance and usefulness of classical results concerning local or non-local efficiency can gather strength by extending to larger regions of neighborhoods; in that way one can investigate where optimality passes into non-optimality if for instance an estimator is locally efficient and non-locally non-efficient. The theory of moderate deviation and Cramer-type large deviation probabilities plays an important role in this context. Examples of the performance of particularly maximum likelihood estimators are presented in $k$-parameter exponential families, a curved exponential family and the double-exponential family.


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Wilbert C. M. Kallenberg. "On Moderate Deviation Theory in Estimation." Ann. Statist. 11 (2) 498 - 504, June, 1983.


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

zbMATH: 0515.62027
MathSciNet: MR696062
Digital Object Identifier: 10.1214/aos/1176346156

Primary: 62F20
Secondary: 60F10 , 62F10

Keywords: First and second order efficiency , maximum likelihood estimator , moderate and Cramer-type large deviations , probability concentration

Rights: Copyright © 1983 Institute of Mathematical Statistics


Vol.11 • No. 2 • June, 1983
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