Bayesian and maximin optimal designs for heteroscedastic multi-factor regression models

In this paper we mainly investigate the problem of optimal designs for multi-factor regression models with partially known heteroscedastic structure. The Bayesian Phi(q)-optimality criterion proposed by Dette and Wong (Ann Stat 24:2108-2127, 1996), which closely resembles Kiefer's Phi(k)-class of criteria, and the standardized maximin D-optimal criterion are considered. More precisely, for heteroscedastic Kronecker product models, it is shown that the product designs formed from optimal designs for sub-models with a single factor are optimal under the two robust criteria. For additive models with intercept, however, sufficient conditions are given in order to search for Bayesian Phi(q)-optimal and standardized maximin D-optimal product designs. Finally, several examples are presented to illustrate the obtained theoretical results.