Robustness of optimal designs for correlated random variables (Q947627)

Suppose that \(Y = (Y_i)\) is a normal random vector with mean \(Xb\) and covariance \(\sigma^2 I_n\), where \(b\) is a \(p\)-dimensional vector \((b_j)\), \(X = (X_{ij})\) is an \(n\times p\) matrix with \(X_{ij}\in[-1, 1]\); this corresponds to a factorial design with \(-1, 1\) representing a low or high level, respectively, or corresponds to a weighing design with \(-1, 1\) representing an object \(j\) with weight \(b_j\) 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 \(Y_i\), \(Y_{i'}\) is \(\rho > 0\) for \(i\neq i'\). 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.