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