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June, 1986 Large Deviations of Estimators
A. D. M. Kester, W. C. M. Kallenberg
Ann. Statist. 14(2): 648-664 (June, 1986). DOI: 10.1214/aos/1176349944


The performance of a sequence of estimators $\{T_n\}$ of $g(\theta)$ can be measured by its inaccuracy rate $-\lim \inf_{n\rightarrow\infty} n^{-1} \log \mathbb{P}_\theta(\|T_n - g(\theta)\| > \varepsilon)$. For fixed $\varepsilon > 0$ optimality of consistent estimators $\operatorname{wrt}$ the inaccuracy rate is investigated. It is shown that for exponential families in standard representation with a convex parameter space the maximum likelihood estimator is optimal. If the parameter space is not convex, which occurs for instance in curved exponential families, in general no optimal estimator exists. For the location problem the inaccuracy rate of $M$-estimators is established. If the underlying density is sufficiently smooth an optimal $M$-estimator is obtained within the class of translation equivariant estimators. Tail-behaviour of location estimators is studied. A connection is made between gross error and inaccuracy rate optimality.


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A. D. M. Kester. W. C. M. Kallenberg. "Large Deviations of Estimators." Ann. Statist. 14 (2) 648 - 664, June, 1986.


Published: June, 1986
First available in Project Euclid: 12 April 2007

zbMATH: 0603.62028
MathSciNet: MR840520
Digital Object Identifier: 10.1214/aos/1176349944

Primary: 62F10
Secondary: 60F10

Rights: Copyright © 1986 Institute of Mathematical Statistics


Vol.14 • No. 2 • June, 1986
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