Efficiency Bounds for Product Designs in Linear Models

We provide lower efficiency bounds for the best product design for an additive multifactor linear model. The A -optimality criterion is used to demonstrate that out bounds are better than the conventional bounds. Applications to other criteria, such as IMSE (integrated mean squared error) criterion are also indicated. In all the cases, the best product design appears to perform better when there are more levels in each factor but decreases when more factors are included. Explicit efficiency formulas for non-additive models are also constructed.