Robustness of optimal designs for correlated random variables

Suppose that Y = (Yi) is a normal random vector with mean Xb and covariance σ2In, where b is a p-dimensional vector (bj), X = (Xij) is an n × p matrix with Xij ∈ {−1, 1}; this corresponds to a factorial design with −1, 1 representing low or high level respectively, or corresponds to a weighing design with −1, 1 representing an object j with weight bj placed on the left and right of a chemical balance respectively. E-optimal designs Z are chosen that are robust in the sense that they remain E-optimal when the covariance of Yi, Yi′ is ρ > 0 for i ≠ i′. Within a smaller class of designs similar results are obtained with respect to a general class of optimality criteria which include the A- and D-criteria.