The Annals of Statistics

On Moderate Deviation Theory in Estimation

Wilbert C. M. Kallenberg

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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.

Article information

Ann. Statist., Volume 11, Number 2 (1983), 498-504.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F20
Secondary: 62F10: Point estimation 60F10: Large deviations

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


Kallenberg, Wilbert C. M. On Moderate Deviation Theory in Estimation. Ann. Statist. 11 (1983), no. 2, 498--504. doi:10.1214/aos/1176346156.

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