# Admissible invariant estimators in a linear model

Kybernetika (2014)

• Volume: 50, Issue: 3, page 310-321
• ISSN: 0023-5954

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## Abstract

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Let $𝐲$ be observation vector in the usual linear model with expectation $𝐀\beta$ and covariance matrix known up to a multiplicative scalar, possibly singular. A linear statistic ${𝐚}^{T}𝐲$ is called invariant estimator for a parametric function $\phi ={𝐜}^{T}\beta$ if its MSE depends on $\beta$ only through $\phi$. It is shown that ${𝐚}^{T}𝐲$ is admissible invariant for $\phi$, if and only if, it is a BLUE of $\phi ,$ in the case when $\phi$ is estimable with zero variance, and it is of the form $k\stackrel{^}{\phi }$, where $k\in 〈0,1〉$ and $\stackrel{^}{\phi }$ is an arbitrary BLUE, otherwise. This result is used in the one- and two-way ANOVA models. Our paper is self-contained and accessible, also for non-specialists.

## How to cite

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Stępniak, Czesław. "Admissible invariant estimators in a linear model." Kybernetika 50.3 (2014): 310-321. <http://eudml.org/doc/261926>.

@article{Stępniak2014,
abstract = {Let $\mathbf \{y\}$ be observation vector in the usual linear model with expectation $\mathbf \{A\beta \}$ and covariance matrix known up to a multiplicative scalar, possibly singular. A linear statistic $\mathbf \{a\}^\{T\} \mathbf \{y\}$ is called invariant estimator for a parametric function $\phi = \mathbf \{c\}^\{T\}\mathbf \{\beta \}$ if its MSE depends on $\mathbf \{\beta \}$ only through $\phi$. It is shown that $\mathbf \{a\}^\{T\}\mathbf \{y\}$ is admissible invariant for $\phi$, if and only if, it is a BLUE of $\phi ,$ in the case when $\phi$ is estimable with zero variance, and it is of the form $k\widehat\{\phi \}$, where $k\in \left\langle 0,1\right\rangle$ and $\widehat\{\phi \}$ is an arbitrary BLUE, otherwise. This result is used in the one- and two-way ANOVA models. Our paper is self-contained and accessible, also for non-specialists.},
author = {Stępniak, Czesław},
journal = {Kybernetika},
keywords = {linear estimator; invariant estimator; admissibility; one-way/two-way ANOVA; linear estimator; invariant estimator; admissibility; one-way/two-way ANOVA},
language = {eng},
number = {3},
pages = {310-321},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Admissible invariant estimators in a linear model},
url = {http://eudml.org/doc/261926},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Stępniak, Czesław
TI - Admissible invariant estimators in a linear model
JO - Kybernetika
PY - 2014
PB - Institute of Information Theory and Automation AS CR
VL - 50
IS - 3
SP - 310
EP - 321
AB - Let $\mathbf {y}$ be observation vector in the usual linear model with expectation $\mathbf {A\beta }$ and covariance matrix known up to a multiplicative scalar, possibly singular. A linear statistic $\mathbf {a}^{T} \mathbf {y}$ is called invariant estimator for a parametric function $\phi = \mathbf {c}^{T}\mathbf {\beta }$ if its MSE depends on $\mathbf {\beta }$ only through $\phi$. It is shown that $\mathbf {a}^{T}\mathbf {y}$ is admissible invariant for $\phi$, if and only if, it is a BLUE of $\phi ,$ in the case when $\phi$ is estimable with zero variance, and it is of the form $k\widehat{\phi }$, where $k\in \left\langle 0,1\right\rangle$ and $\widehat{\phi }$ is an arbitrary BLUE, otherwise. This result is used in the one- and two-way ANOVA models. Our paper is self-contained and accessible, also for non-specialists.
LA - eng
KW - linear estimator; invariant estimator; admissibility; one-way/two-way ANOVA; linear estimator; invariant estimator; admissibility; one-way/two-way ANOVA
UR - http://eudml.org/doc/261926
ER -

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