Kybernetika 50 no. 3, 310-321, 2014

Admissible invariant estimators in a linear model

Czesław StępniakDOI: 10.14736/kyb-2014-3-0310


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.


admissibility, linear estimator, invariant estimator, one-way/two-way ANOVA


62J05, 62C05


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