D-optimal designs for correlated random vectors

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. Given a family D of D-optimal designs, a design Z in D is chosen that is robust in the sense that Z is D-optimal in D when the components Yi are dependent: for i≠i′, the covariance of Yi,Yi′ is ρ≠0. Such designs Z merely depend on the sign of ρ. The general results are applied to the situation where 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 being placed on the left and right side of a chemical balance, respectively.