Geodesic
Admissible invariant estimators in a linear model
Kybernetika, Tome 50 (2014) no. 3, pp. 310-321
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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.
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.
DOI : 10.14736/kyb-2014-3-0310
Classification : 62C05, 62C15, 62J05, 62J10
Keywords: linear estimator; invariant estimator; admissibility; one-way/two-way ANOVA 
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Stępniak, Czesław. Admissible invariant estimators in a linear model. Kybernetika, Tome 50 (2014) no. 3, pp. 310-321. doi: 10.14736/kyb-2014-3-0310

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