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March, 1983 Invariantly Sufficient Equivariant Statistics and Characterizations of Normality in Translation Classes
W. Eberl Jr
Ann. Statist. 11(1): 330-336 (March, 1983). DOI: 10.1214/aos/1176346084

Abstract

It is shown that an equivariant statistic $S$ is invariantly sufficient iff the generated $\sigma$-algebra and the $\sigma$-algebra of the invariant Borel sets are independent, and that if $S$ is invariantly sufficient and equivariant, then the Pitman estimator for location parameter $\gamma$ is given by $S - E_0(S)$. For independent $X_1, \cdots, X_n$, the existence of an invariantly sufficient equivariant linear statistic is characterized by the normality of $X_1, \cdots, X_n$. Then, the independence of $X_1, \cdots, X_n$ is replaced by a linear framework in which there are established characterizations of the normality of $X = (X_1, \cdots, X_n)$ by properties (invariant sufficiency, admissibility, optimality) of the minimum variance unbiased linear estimator for $\gamma$.

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W. Eberl Jr. "Invariantly Sufficient Equivariant Statistics and Characterizations of Normality in Translation Classes." Ann. Statist. 11 (1) 330 - 336, March, 1983. https://doi.org/10.1214/aos/1176346084

Information

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

zbMATH: 0533.62006
MathSciNet: MR684891
Digital Object Identifier: 10.1214/aos/1176346084

Subjects:
Primary: 62B05
Secondary: 62C15 , 62E10 , 62G05

Keywords: Admissibility , characterizations of normality in translation classes by statistical properties , characterizations of the MVU linear estimator in linear processes , Equivariance , Invariance , invariant sufficiency , normality in translation classes , Pitman estimator

Rights: Copyright © 1983 Institute of Mathematical Statistics

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Vol.11 • No. 1 • March, 1983
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