Comparison of Linear Experiments with Known Covariances

Abstract

For two linear experiments $d_1 = L(X_1\beta, V_1)$ and $d_2 = L(X_2\beta, V_2)$ where the covariances $V_1$ and $V_2$ are known and can be singular or nonsingular, we characterize the following relations: $d_1$ at least as good as $d_2, d_1$ better than $d_2$, and $d_1$ equivalent to $d_2$. Sometimes only a subset of parameters is of interest to the experimenter. We extend the above relations between $d_1$ and $d_2$ to estimation of a common subset of parameters and give analogous characterizations. Three examples are given.