The Annals of Mathematical Statistics

A Unified Theory of Estimation, I

Allan Birnbaum

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This paper extends and unifies some previous formulations and theories of estimation for one-parameter problems. The basic criterion used is admissibility of a point estimator, defined with reference to its full distribution rather than special loss functions such as squared error. Theoretical methods of characterizing admissible estimators are given, and practical computational methods for their use are illustrated. Point, confidence limit, and confidence interval estimation are included in a single theoretical formulation, and incorporated into estimators of an "omnibus" form called "confidence curves." The usefulness of the latter for some applications as well as theoretical purposes is illustrated. Fisher's maximum likelihood principle of estimation is generalized, given exact (non-asymptotic) justification, and unified with the theory of tests and confidence regions of Neyman and Pearson. Relations between exact and asymptotic results are discussed. Further developments, including multiparameter and nuisance parameter problems, problems of choice among admissible estimators, formal and informal criteria for optimality, and related problems in the foundations of statistical inference, will be presented subsequently.

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Ann. Math. Statist. Volume 32, Number 1 (1961), 112-135.

First available in Project Euclid: 27 April 2007

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Birnbaum, Allan. A Unified Theory of Estimation, I. Ann. Math. Statist. 32 (1961), no. 1, 112--135. doi:10.1214/aoms/1177705145.

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