Combining Experiments Under Gauss-Markov Models
Ahmadreza Hedayat, Dibyen Majumdar · Journal of the American Statistical Association · 1985
Consider two experiments with homoscedastic models E(Yi ) = Xi θ + Zi ψ, V(Yi ) = σ2 Ini (i = 1, 2). The matrix gi = X'i (Ini — Zi (Z'iZi )Xi can be called the information matrix of linear estimable functions of θ for experiment i. The information matrix g for the combined data (Y 1, Y 2) may be similarly defined. A proof of g ⩾ g 1 + g 2 (⩾ in the sense of nonnegative definiteness) is given, and equality is interpreted. g = g 1 + g 2 if and only if all best linear unbiased estimators (BLUE's) from the combined experiment are linear combinations of BLUE's from the original experiments. Some characterizations of design matrices are obtained for which the BLUE of p'θ from the combined experiment is a linear combination of the BLUE's of p'θ from the individual experiments, for all p'θ that are estimable in both individual experiments.