D-optimal designs for correlated random vectors

Abstract

Suppose that Y = ( Y i ) is a normal random vector with mean Xb and covariance σ 2 I n , where b is a p -dimensional vector ( b j ) , X = ( X ij ) 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 Y i are dependent: for i ≠ i ′ , the covariance of Y i , Y i ′ is ρ ≠ 0 . Such designs Z merely depend on the sign of ρ . The general results are applied to the situation where X ij ∈ { - 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 b j being placed on the left and right side of a chemical balance, respectively.