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Invariance, Haar Measures, and Equivariant Estimators

  • Christian P. Robert
Part of the Springer Texts in Statistics book series (STS)

Abstract

Invariance can be seen as a notion introduced in frequentist settings to restrict the range of acceptable estimators sufficiently so that an optimal estimator can be derived. From this point of view, it appears as an alternative to unbiasedness and is thus similarly at odds with the Bayesian paradigm. However, it also partakes from a non-decision-theoretic heuristic, namely that estimators should meet some consistency requirements under a group of transformations, and it is thus of interest to consider this notion. Moreover, a Bayesian perspective of invariance is justified by the fact that optimal (equivariant) estimators are always Bayes or generalized Bayes estimators. The corresponding measures can be considered as noninformative priors induced by the invariance structure. Therefore, a Bayesian study of invariance is appealling, not because classical optimality once more relies on Bayesian estimators, but mainly because of the connection between invariance structures and the derivation of noninformative distributions.

Keywords

Maximum Likelihood Estimator Location Parameter Haar Measure Invariance Structure Quadratic Loss 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Christian P. Robert
    • 1
  1. 1.URA CNRS 1378 — Dépt. de Math.Université de RouenMont Saint Aignan CedexFrance

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