Comparison of Linear Normal Experiments

Consider independent and normally distributed random variables X₁,⋯, Xₙ such that 0 < Var Xᵢ = σ²; i = 1,⋯, n and E(X₁,⋯, Xₙ)' = A'β where $A'$ is a known n × k matrix and β = (β₁,⋯, βₖ)' is an unknown column matrix. (The prime denotes transposition.) The cases of known and totally unknown σ² are considered simultaneously. Denote the experiment obtained by observing X₁,⋯, Xₙ by EA. Let A and B be matrices of, respectively, dimensions nA × k and nB × k. Then, if σ² is known, (if σ² is unknown) EA is more informative than EB if and only if $AA' - BB'$ is nonnegative definite (and nA nB + rank (AA' - BB')).